L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s − 25·5-s + 36·6-s − 10.5·7-s − 64·8-s + 81·9-s + 100·10-s + 66.1·11-s − 144·12-s − 874.·13-s + 42.0·14-s + 225·15-s + 256·16-s + 1.22e3·17-s − 324·18-s + 361·19-s − 400·20-s + 94.6·21-s − 264.·22-s − 3.27e3·23-s + 576·24-s + 625·25-s + 3.49e3·26-s − 729·27-s − 168.·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s − 0.0811·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.164·11-s − 0.288·12-s − 1.43·13-s + 0.0573·14-s + 0.258·15-s + 0.250·16-s + 1.03·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.0468·21-s − 0.116·22-s − 1.28·23-s + 0.204·24-s + 0.200·25-s + 1.01·26-s − 0.192·27-s − 0.0405·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.6084479929\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6084479929\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 + 9T \) |
| 5 | \( 1 + 25T \) |
| 19 | \( 1 - 361T \) |
good | 7 | \( 1 + 10.5T + 1.68e4T^{2} \) |
| 11 | \( 1 - 66.1T + 1.61e5T^{2} \) |
| 13 | \( 1 + 874.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.22e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.27e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.08e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.22e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.93e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.27e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.47e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.73e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.02e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.59e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.19e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.23e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.34e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.59e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.98e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.03e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.96e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.98e3T + 8.58e9T^{2} \) |
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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
−9.911158841976118861973721075362, −9.308838829658391643315891885351, −7.929366253079972514727553207492, −7.52498081836296588534832198466, −6.43981667502678933364910796415, −5.47200814550344151890154506355, −4.37262144756782869309085150541, −3.08317403865466998866041715502, −1.74707286055795662914597954084, −0.43353352370111677893852619581,
0.43353352370111677893852619581, 1.74707286055795662914597954084, 3.08317403865466998866041715502, 4.37262144756782869309085150541, 5.47200814550344151890154506355, 6.43981667502678933364910796415, 7.52498081836296588534832198466, 7.929366253079972514727553207492, 9.308838829658391643315891885351, 9.911158841976118861973721075362