L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s − 21.6·5-s − 36·6-s − 77.1·7-s + 64·8-s + 81·9-s − 86.6·10-s − 128.·11-s − 144·12-s + 503.·13-s − 308.·14-s + 194.·15-s + 256·16-s + 1.61e3·17-s + 324·18-s + 2.07e3·19-s − 346.·20-s + 694.·21-s − 513.·22-s − 4.00e3·23-s − 576·24-s − 2.65e3·25-s + 2.01e3·26-s − 729·27-s − 1.23e3·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.387·5-s − 0.408·6-s − 0.595·7-s + 0.353·8-s + 0.333·9-s − 0.273·10-s − 0.320·11-s − 0.288·12-s + 0.827·13-s − 0.421·14-s + 0.223·15-s + 0.250·16-s + 1.35·17-s + 0.235·18-s + 1.31·19-s − 0.193·20-s + 0.343·21-s − 0.226·22-s − 1.58·23-s − 0.204·24-s − 0.849·25-s + 0.584·26-s − 0.192·27-s − 0.297·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 + 9T \) |
| 59 | \( 1 - 3.48e3T \) |
good | 5 | \( 1 + 21.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + 77.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + 128.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 503.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.61e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.07e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.00e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.40e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.18e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.09e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.54e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.65e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.13e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.77e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 615.T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.17e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.85e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.05e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.86e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.33e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.00e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.20e4T + 8.58e9T^{2} \) |
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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
−10.31304224039756138214784719164, −9.511725370060531149226546699159, −7.979706212586706409996223264644, −7.24628900911554681465813258170, −5.95372375159405436978945607175, −5.47728576093123825039411577688, −4.00291831858579432580525572815, −3.25332684359677063783776366608, −1.52968620266401120311138655911, 0,
1.52968620266401120311138655911, 3.25332684359677063783776366608, 4.00291831858579432580525572815, 5.47728576093123825039411577688, 5.95372375159405436978945607175, 7.24628900911554681465813258170, 7.979706212586706409996223264644, 9.511725370060531149226546699159, 10.31304224039756138214784719164