L(s) = 1 | − 6·3-s − 9·5-s + 27·9-s − 5·11-s + 19·13-s + 54·15-s + 4·17-s + 117·19-s − 148·23-s − 63·25-s − 108·27-s − 95·29-s − 360·31-s + 30·33-s − 53·37-s − 114·39-s + 170·41-s − 403·43-s − 243·45-s + 368·47-s − 24·51-s + 697·53-s + 45·55-s − 702·57-s − 585·59-s + 1.16e3·61-s − 171·65-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.804·5-s + 9-s − 0.137·11-s + 0.405·13-s + 0.929·15-s + 0.0570·17-s + 1.41·19-s − 1.34·23-s − 0.503·25-s − 0.769·27-s − 0.608·29-s − 2.08·31-s + 0.158·33-s − 0.235·37-s − 0.468·39-s + 0.647·41-s − 1.42·43-s − 0.804·45-s + 1.14·47-s − 0.0658·51-s + 1.80·53-s + 0.110·55-s − 1.63·57-s − 1.29·59-s + 2.43·61-s − 0.326·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | | \( 1 \) |
good | 5 | $D_{4}$ | \( 1 + 9 T + 144 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T - 488 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 19 T + 4358 T^{2} - 19 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 7810 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 117 T + 10954 T^{2} - 117 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 148 T + 27790 T^{2} + 148 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 95 T + 47878 T^{2} + 95 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 360 T + 91477 T^{2} + 360 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 53 T + 101882 T^{2} + 53 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 170 T + 104162 T^{2} - 170 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 403 T + 107580 T^{2} + 403 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 368 T + 77882 T^{2} - 368 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 697 T + 7850 p T^{2} - 697 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 585 T + 404278 T^{2} + 585 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1160 T + 691382 T^{2} - 1160 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 233 T - 57688 T^{2} + 233 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 616 T + 81466 T^{2} + 616 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 817 T + 443820 T^{2} - 817 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 802 T + 855999 T^{2} - 802 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 283 T + 596860 T^{2} + 283 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1858 T + 2211874 T^{2} - 1858 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1729 T + 2190800 T^{2} + 1729 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.232402943816877004546267887345, −8.084294683307314176167441609457, −7.46705679693350887609067261256, −7.38605384484623317024529394333, −6.96273980807054071107151762700, −6.49949924546603213422567137501, −5.98601650598892647282687477334, −5.65176391001773786413303138164, −5.23391041393623504303682546275, −5.19265937773197654925287430332, −4.17419845927221515075399876332, −4.17223304841035422242391495560, −3.49025522646546413279986908319, −3.46292562163118830031119333370, −2.40004342483679636100932789409, −2.02701624497428652809097063538, −1.32164763065636438092304629326, −0.864298108556478629796836981150, 0, 0,
0.864298108556478629796836981150, 1.32164763065636438092304629326, 2.02701624497428652809097063538, 2.40004342483679636100932789409, 3.46292562163118830031119333370, 3.49025522646546413279986908319, 4.17223304841035422242391495560, 4.17419845927221515075399876332, 5.19265937773197654925287430332, 5.23391041393623504303682546275, 5.65176391001773786413303138164, 5.98601650598892647282687477334, 6.49949924546603213422567137501, 6.96273980807054071107151762700, 7.38605384484623317024529394333, 7.46705679693350887609067261256, 8.084294683307314176167441609457, 8.232402943816877004546267887345