| L(s) = 1 | − 6.54e5·2-s − 6.70e11·4-s + 7.97e16·7-s + 1.15e18·8-s + 1.21e19·9-s + 9.76e20·11-s − 5.22e22·14-s − 2.13e22·16-s − 7.96e24·18-s − 6.39e26·22-s − 6.38e26·23-s + 9.09e27·25-s − 5.35e28·28-s + 1.32e29·29-s − 1.26e30·32-s − 8.15e30·36-s − 9.64e30·37-s − 5.77e32·43-s − 6.55e32·44-s + 4.17e32·46-s + 6.36e33·49-s − 5.95e33·50-s + 2.59e34·53-s + 9.24e34·56-s − 8.65e34·58-s + 9.70e35·63-s + 8.48e35·64-s + ⋯ |
| L(s) = 1 | − 0.624·2-s − 0.610·4-s + 7-s + 1.00·8-s + 9-s + 1.45·11-s − 0.624·14-s − 0.0176·16-s − 0.624·18-s − 0.906·22-s − 0.371·23-s + 25-s − 0.610·28-s + 0.747·29-s − 0.994·32-s − 0.610·36-s − 0.417·37-s − 1.23·43-s − 0.885·44-s + 0.232·46-s + 49-s − 0.624·50-s + 0.848·53-s + 1.00·56-s − 0.466·58-s + 63-s + 0.638·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(41-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+20) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{41}{2})\) |
\(\approx\) |
\(1.902119758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.902119758\) |
| \(L(21)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - p^{20} T \) |
| good | 2 | \( 1 + 654751 T + p^{40} T^{2} \) |
| 3 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 5 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 11 | \( 1 - \)\(97\!\cdots\!74\)\( T + p^{40} T^{2} \) |
| 13 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 17 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 19 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 23 | \( 1 + \)\(63\!\cdots\!26\)\( T + p^{40} T^{2} \) |
| 29 | \( 1 - \)\(13\!\cdots\!74\)\( T + p^{40} T^{2} \) |
| 31 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 37 | \( 1 + \)\(96\!\cdots\!26\)\( T + p^{40} T^{2} \) |
| 41 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 43 | \( 1 + \)\(57\!\cdots\!26\)\( T + p^{40} T^{2} \) |
| 47 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 53 | \( 1 - \)\(25\!\cdots\!02\)\( T + p^{40} T^{2} \) |
| 59 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 61 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 67 | \( 1 + \)\(59\!\cdots\!26\)\( T + p^{40} T^{2} \) |
| 71 | \( 1 - \)\(20\!\cdots\!74\)\( T + p^{40} T^{2} \) |
| 73 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 79 | \( 1 - \)\(17\!\cdots\!74\)\( T + p^{40} T^{2} \) |
| 83 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 89 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 97 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.68909142621842487746815716435, −12.10953552442318558045580719230, −10.57438922334354711884799587529, −9.321548049094323142157179960694, −8.220970746392455765559409572788, −6.87271437510415180563381429724, −4.89329654923641429222853625035, −3.93873307073985493207595473400, −1.67249784307047529416145631203, −0.894353921675196380085577415088,
0.894353921675196380085577415088, 1.67249784307047529416145631203, 3.93873307073985493207595473400, 4.89329654923641429222853625035, 6.87271437510415180563381429724, 8.220970746392455765559409572788, 9.321548049094323142157179960694, 10.57438922334354711884799587529, 12.10953552442318558045580719230, 13.68909142621842487746815716435
