| L(s) = 1 | + 340·4-s − 1.14e3·5-s − 1.09e5·11-s + 2.90e5·16-s − 6.36e5·19-s − 3.87e5·20-s − 1.91e4·25-s + 3.53e6·29-s − 1.05e7·31-s + 1.67e7·41-s − 3.73e7·44-s + 5.72e7·49-s + 1.25e8·55-s + 4.60e8·59-s + 3.60e8·61-s + 2.47e8·64-s + 4.76e7·71-s − 2.16e8·76-s − 7.28e8·79-s − 3.31e8·80-s + 1.58e9·89-s + 7.26e8·95-s − 6.51e6·100-s − 2.27e9·101-s − 2.13e9·109-s + 1.20e9·116-s − 1.29e9·121-s + ⋯ |
| L(s) = 1 | + 0.664·4-s − 0.815·5-s − 2.26·11-s + 1.10·16-s − 1.12·19-s − 0.541·20-s − 0.00980·25-s + 0.927·29-s − 2.05·31-s + 0.927·41-s − 1.50·44-s + 1.41·49-s + 1.84·55-s + 4.95·59-s + 3.33·61-s + 1.84·64-s + 0.222·71-s − 0.744·76-s − 2.10·79-s − 0.903·80-s + 2.67·89-s + 0.914·95-s − 0.00651·100-s − 2.17·101-s − 1.44·109-s + 0.615·116-s − 0.547·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.393795304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.393795304\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $D_{4}$ | \( 1 + 228 p T + 422 p^{5} T^{2} + 228 p^{10} T^{3} + p^{18} T^{4} \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 - 85 p^{2} T^{2} - 2733 p^{6} T^{4} - 85 p^{20} T^{6} + p^{36} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - 166900 p^{3} T^{2} + 1461117678198 p^{4} T^{4} - 166900 p^{21} T^{6} + p^{36} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 54984 T + 5180465446 T^{2} + 54984 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 35613791860 T^{2} + \)\(53\!\cdots\!58\)\( T^{4} - 35613791860 p^{18} T^{6} + p^{36} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 285780369220 T^{2} + \)\(48\!\cdots\!18\)\( T^{4} - 285780369220 p^{18} T^{6} + p^{36} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 16760 p T + 505756418358 T^{2} + 16760 p^{10} T^{3} + p^{18} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 5779790962540 T^{2} + \)\(14\!\cdots\!38\)\( T^{4} - 5779790962540 p^{18} T^{6} + p^{36} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 1765860 T + 26950935551038 T^{2} - 1765860 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 5293856 T + 59464921598526 T^{2} + 5293856 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 231603274936660 T^{2} + \)\(44\!\cdots\!58\)\( T^{4} - 231603274936660 p^{18} T^{6} + p^{36} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 8394276 T + 221313076168966 T^{2} - 8394276 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 614109141147100 T^{2} + \)\(57\!\cdots\!98\)\( T^{4} - 614109141147100 p^{18} T^{6} + p^{36} T^{8} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 1368976020813580 T^{2} + \)\(29\!\cdots\!78\)\( T^{4} - 1368976020813580 p^{18} T^{6} + p^{36} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 7684297973864980 T^{2} + \)\(36\!\cdots\!78\)\( T^{4} - 7684297973864980 p^{18} T^{6} + p^{36} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 230414520 T + 28555631923987078 T^{2} - 230414520 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 180245284 T + 30154717014478446 T^{2} - 180245284 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 41160407446058180 T^{2} + \)\(18\!\cdots\!18\)\( T^{4} + 41160407446058180 p^{18} T^{6} + p^{36} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 23805936 T + 85782754020107086 T^{2} - 23805936 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 229489314868712740 T^{2} + \)\(37\!\cdots\!22\)\( p^{2} T^{4} - 229489314868712740 p^{18} T^{6} + p^{36} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 364021760 T + 220545463862625438 T^{2} + 364021760 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 434569632367965820 T^{2} + \)\(10\!\cdots\!18\)\( T^{4} - 434569632367965820 p^{18} T^{6} + p^{36} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 791350380 T + 832192702699668118 T^{2} - 791350380 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 2561123777205326980 T^{2} + \)\(27\!\cdots\!78\)\( T^{4} - 2561123777205326980 p^{18} T^{6} + p^{36} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.01683502726741744639455416897, −9.503847103367752762416587986939, −9.084378023060380496369935654272, −8.540856198010319033956457532295, −8.366538126190326081450348427014, −8.138615209343430532338855799889, −7.84695280548816445593486160554, −7.35187250360813660435780509612, −7.17568885638856124330146944124, −6.76839267446191538850381642785, −6.52646129821853441478236262035, −5.74411179177881503468225659282, −5.49157840044399043533756646601, −5.32790441163938011720935577037, −5.03399792452507273727433373013, −4.07647741057103887809356520821, −3.97000821875881481621849033512, −3.74729646524425962088778689553, −2.91191379957126499927687700895, −2.65893894080984755902079553637, −2.12796389799797064857445395852, −2.09992988791923804794318287799, −1.07176890370784029230433237863, −0.66820303089539268786698504141, −0.31429343395810081112735167126,
0.31429343395810081112735167126, 0.66820303089539268786698504141, 1.07176890370784029230433237863, 2.09992988791923804794318287799, 2.12796389799797064857445395852, 2.65893894080984755902079553637, 2.91191379957126499927687700895, 3.74729646524425962088778689553, 3.97000821875881481621849033512, 4.07647741057103887809356520821, 5.03399792452507273727433373013, 5.32790441163938011720935577037, 5.49157840044399043533756646601, 5.74411179177881503468225659282, 6.52646129821853441478236262035, 6.76839267446191538850381642785, 7.17568885638856124330146944124, 7.35187250360813660435780509612, 7.84695280548816445593486160554, 8.138615209343430532338855799889, 8.366538126190326081450348427014, 8.540856198010319033956457532295, 9.084378023060380496369935654272, 9.503847103367752762416587986939, 10.01683502726741744639455416897
