L(s) = 1 | − 5·2-s − 103·4-s − 930·7-s + 1.15e3·8-s + 8.45e3·11-s − 6.22e3·13-s + 4.65e3·14-s + 7.40e3·16-s − 9.59e3·17-s − 4.58e4·19-s − 4.22e4·22-s − 1.02e5·23-s + 3.11e4·26-s + 9.57e4·28-s + 8.75e4·29-s − 7.62e4·31-s − 1.84e5·32-s + 4.79e4·34-s − 2.64e5·37-s + 2.29e5·38-s + 1.03e5·41-s + 3.24e5·43-s − 8.70e5·44-s + 5.10e5·46-s + 8.55e5·47-s + 4.13e4·49-s + 6.40e5·52-s + ⋯ |
L(s) = 1 | − 0.441·2-s − 0.804·4-s − 1.02·7-s + 0.797·8-s + 1.91·11-s − 0.785·13-s + 0.452·14-s + 0.452·16-s − 0.473·17-s − 1.53·19-s − 0.845·22-s − 1.75·23-s + 0.347·26-s + 0.824·28-s + 0.666·29-s − 0.459·31-s − 0.997·32-s + 0.209·34-s − 0.858·37-s + 0.678·38-s + 0.234·41-s + 0.622·43-s − 1.54·44-s + 0.773·46-s + 1.20·47-s + 0.0502·49-s + 0.631·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.7380388185\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7380388185\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 5 T + p^{7} T^{2} \) |
| 7 | \( 1 + 930 T + p^{7} T^{2} \) |
| 11 | \( 1 - 8450 T + p^{7} T^{2} \) |
| 13 | \( 1 + 6220 T + p^{7} T^{2} \) |
| 17 | \( 1 + 9590 T + p^{7} T^{2} \) |
| 19 | \( 1 + 45884 T + p^{7} T^{2} \) |
| 23 | \( 1 + 4440 p T + p^{7} T^{2} \) |
| 29 | \( 1 - 87550 T + p^{7} T^{2} \) |
| 31 | \( 1 + 76212 T + p^{7} T^{2} \) |
| 37 | \( 1 + 264440 T + p^{7} T^{2} \) |
| 41 | \( 1 - 103600 T + p^{7} T^{2} \) |
| 43 | \( 1 - 324680 T + p^{7} T^{2} \) |
| 47 | \( 1 - 855880 T + p^{7} T^{2} \) |
| 53 | \( 1 + 958190 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1239550 T + p^{7} T^{2} \) |
| 61 | \( 1 - 628522 T + p^{7} T^{2} \) |
| 67 | \( 1 + 310380 T + p^{7} T^{2} \) |
| 71 | \( 1 - 3934300 T + p^{7} T^{2} \) |
| 73 | \( 1 - 4556090 T + p^{7} T^{2} \) |
| 79 | \( 1 - 5371644 T + p^{7} T^{2} \) |
| 83 | \( 1 + 6711060 T + p^{7} T^{2} \) |
| 89 | \( 1 - 3346500 T + p^{7} T^{2} \) |
| 97 | \( 1 + 15829730 T + p^{7} T^{2} \) |
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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
−10.73353993538823877838355873177, −9.733661888702421367735472831931, −9.188657806569647718208339672327, −8.267778526699003535558187127381, −6.90018106884949522900594907912, −6.07740484898910050461792700199, −4.44137443041451709694477958287, −3.72384734812860435034757589473, −1.95076330941632227941950954174, −0.47414023205130675160685824599,
0.47414023205130675160685824599, 1.95076330941632227941950954174, 3.72384734812860435034757589473, 4.44137443041451709694477958287, 6.07740484898910050461792700199, 6.90018106884949522900594907912, 8.267778526699003535558187127381, 9.188657806569647718208339672327, 9.733661888702421367735472831931, 10.73353993538823877838355873177