| L(s) = 1 | + 7.78·2-s − 451.·4-s − 625·5-s + 1.83e3·7-s − 7.50e3·8-s − 4.86e3·10-s − 4.43e4·11-s + 1.36e5·13-s + 1.43e4·14-s + 1.72e5·16-s + 2.53e5·17-s + 8.54e4·19-s + 2.82e5·20-s − 3.45e5·22-s + 9.79e5·23-s + 3.90e5·25-s + 1.06e6·26-s − 8.30e5·28-s − 2.58e6·29-s + 8.94e6·31-s + 5.18e6·32-s + 1.97e6·34-s − 1.14e6·35-s + 1.56e7·37-s + 6.65e5·38-s + 4.68e6·40-s − 2.44e7·41-s + ⋯ |
| L(s) = 1 | + 0.344·2-s − 0.881·4-s − 0.447·5-s + 0.289·7-s − 0.647·8-s − 0.153·10-s − 0.914·11-s + 1.32·13-s + 0.0996·14-s + 0.658·16-s + 0.736·17-s + 0.150·19-s + 0.394·20-s − 0.314·22-s + 0.729·23-s + 0.200·25-s + 0.456·26-s − 0.255·28-s − 0.679·29-s + 1.74·31-s + 0.874·32-s + 0.253·34-s − 0.129·35-s + 1.36·37-s + 0.0517·38-s + 0.289·40-s − 1.35·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.638672895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.638672895\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 625T \) |
| good | 2 | \( 1 - 7.78T + 512T^{2} \) |
| 7 | \( 1 - 1.83e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.43e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.36e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.53e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.54e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 9.79e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.58e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.94e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.56e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.44e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.27e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 6.16e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.70e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 8.35e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.48e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.68e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.10e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.43e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.55e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.55e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.24e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.19e9T + 7.60e17T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.72342347554983072557557268868, −12.91364408409424783332083271215, −11.60856379249352492817263791990, −10.28302127046966178966837884250, −8.812280147862906492745604514838, −7.80400228166169182857021768609, −5.85371723475241735936610684749, −4.56785295714959469379177249519, −3.21509233838646124568201381732, −0.840875712404553090896598392552,
0.840875712404553090896598392552, 3.21509233838646124568201381732, 4.56785295714959469379177249519, 5.85371723475241735936610684749, 7.80400228166169182857021768609, 8.812280147862906492745604514838, 10.28302127046966178966837884250, 11.60856379249352492817263791990, 12.91364408409424783332083271215, 13.72342347554983072557557268868
