| L(s) = 1 | + 12·3-s − 8·4-s + 90·9-s − 96·12-s + 32·13-s + 48·16-s − 96·17-s + 408·23-s + 32·25-s + 540·27-s + 504·29-s − 720·36-s + 384·39-s + 944·43-s + 576·48-s + 148·49-s − 1.15e3·51-s − 256·52-s − 240·53-s + 352·61-s − 256·64-s + 768·68-s + 4.89e3·69-s + 384·75-s − 2.81e3·79-s + 2.83e3·81-s + 6.04e3·87-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 4-s + 10/3·9-s − 2.30·12-s + 0.682·13-s + 3/4·16-s − 1.36·17-s + 3.69·23-s + 0.255·25-s + 3.84·27-s + 3.22·29-s − 3.33·36-s + 1.57·39-s + 3.34·43-s + 1.73·48-s + 0.431·49-s − 3.16·51-s − 0.682·52-s − 0.622·53-s + 0.738·61-s − 1/2·64-s + 1.36·68-s + 8.54·69-s + 0.591·75-s − 4.01·79-s + 35/9·81-s + 7.45·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.982358842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.982358842\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{4} \) |
| 13 | $C_2^2$ | \( 1 - 32 T - 66 p T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 32 T^{2} - 18066 T^{4} - 32 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2840 T^{2} + 4893054 T^{4} - 2840 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 48 T + 4894 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 25060 T^{2} + 249682614 T^{4} - 25060 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 204 T + 29230 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 126 T + p^{3} T^{2} )^{4} \) |
| 31 | $D_4\times C_2$ | \( 1 - 45076 T^{2} + 1944204774 T^{4} - 45076 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 38308 T^{2} + 3099043734 T^{4} - 38308 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 199328 T^{2} + 19409850078 T^{4} - 199328 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 472 T + 126582 T^{2} - 472 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 397688 T^{2} + 61092073806 T^{4} - 397688 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 120 T + 163654 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 294440 T^{2} + 87890204382 T^{4} - 294440 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 176 T + 191814 T^{2} - 176 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 808996 T^{2} + 306146839542 T^{4} - 808996 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 1338152 T^{2} + 703613369166 T^{4} - 1338152 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 1182100 T^{2} + 648010275846 T^{4} - 1182100 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 1408 T + 1129182 T^{2} + 1408 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1771880 T^{2} + 1421063042046 T^{4} - 1771880 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 1092640 T^{2} + 1272102276894 T^{4} + 1092640 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 801932 T^{2} + 1563264286566 T^{4} + 801932 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−10.01212377422176242986762389472, −9.806973770829212696553129896011, −9.323750989861818586197186117409, −9.138956475553040297797610701397, −9.054203271851660746408834791272, −8.671930771738539239974721580530, −8.509671872477591221649617018717, −8.202633295770237959255919130886, −8.066592956108280288593544095572, −7.28053483923749639640084122326, −7.13752015036002490526145740561, −7.04116662271531806678796209340, −6.56660767519579750761053029005, −6.06061718551827081948893105120, −5.60550363508004345003225639190, −5.02026036947979499122918426666, −4.49459280846454133573926399919, −4.49029118411759749765127819547, −4.18483392732049757062800268204, −3.29467084562829624286380309273, −3.22512329705536555716638975578, −2.57249489864379938617451724053, −2.48466401194717000504660827207, −1.10204238607028218128013573929, −1.06754726017078016176209934091,
1.06754726017078016176209934091, 1.10204238607028218128013573929, 2.48466401194717000504660827207, 2.57249489864379938617451724053, 3.22512329705536555716638975578, 3.29467084562829624286380309273, 4.18483392732049757062800268204, 4.49029118411759749765127819547, 4.49459280846454133573926399919, 5.02026036947979499122918426666, 5.60550363508004345003225639190, 6.06061718551827081948893105120, 6.56660767519579750761053029005, 7.04116662271531806678796209340, 7.13752015036002490526145740561, 7.28053483923749639640084122326, 8.066592956108280288593544095572, 8.202633295770237959255919130886, 8.509671872477591221649617018717, 8.671930771738539239974721580530, 9.054203271851660746408834791272, 9.138956475553040297797610701397, 9.323750989861818586197186117409, 9.806973770829212696553129896011, 10.01212377422176242986762389472
