| L(s) = 1 | + 312·5-s − 323·7-s − 3.72e3·11-s − 1.41e4·13-s + 1.59e4·17-s − 2.24e4·19-s − 5.77e4·23-s + 1.92e4·25-s + 1.66e5·29-s − 9.48e4·31-s − 1.00e5·35-s + 4.53e5·37-s + 6.27e5·41-s + 4.24e4·43-s + 1.23e6·47-s − 7.19e5·49-s + 1.07e5·53-s − 1.16e6·55-s + 2.47e6·59-s + 2.87e6·61-s − 4.42e6·65-s − 1.50e6·67-s − 4.73e6·71-s − 8.51e4·73-s + 1.20e6·77-s + 1.18e6·79-s + 1.11e6·83-s + ⋯ |
| L(s) = 1 | + 1.11·5-s − 0.355·7-s − 0.842·11-s − 1.78·13-s + 0.785·17-s − 0.749·19-s − 0.990·23-s + 0.246·25-s + 1.26·29-s − 0.571·31-s − 0.397·35-s + 1.47·37-s + 1.42·41-s + 0.0814·43-s + 1.73·47-s − 0.873·49-s + 0.0989·53-s − 0.940·55-s + 1.57·59-s + 1.62·61-s − 1.99·65-s − 0.609·67-s − 1.56·71-s − 0.0256·73-s + 0.299·77-s + 0.269·79-s + 0.214·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.984546106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.984546106\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 312 T + p^{7} T^{2} \) |
| 7 | \( 1 + 323 T + p^{7} T^{2} \) |
| 11 | \( 1 + 3720 T + p^{7} T^{2} \) |
| 13 | \( 1 + 14179 T + p^{7} T^{2} \) |
| 17 | \( 1 - 936 p T + p^{7} T^{2} \) |
| 19 | \( 1 + 22421 T + p^{7} T^{2} \) |
| 23 | \( 1 + 57768 T + p^{7} T^{2} \) |
| 29 | \( 1 - 166656 T + p^{7} T^{2} \) |
| 31 | \( 1 + 94820 T + p^{7} T^{2} \) |
| 37 | \( 1 - 453971 T + p^{7} T^{2} \) |
| 41 | \( 1 - 627072 T + p^{7} T^{2} \) |
| 43 | \( 1 - 42472 T + p^{7} T^{2} \) |
| 47 | \( 1 - 1235256 T + p^{7} T^{2} \) |
| 53 | \( 1 - 107280 T + p^{7} T^{2} \) |
| 59 | \( 1 - 2479224 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2874383 T + p^{7} T^{2} \) |
| 67 | \( 1 + 1501097 T + p^{7} T^{2} \) |
| 71 | \( 1 + 4733136 T + p^{7} T^{2} \) |
| 73 | \( 1 + 85111 T + p^{7} T^{2} \) |
| 79 | \( 1 - 1180819 T + p^{7} T^{2} \) |
| 83 | \( 1 - 1116528 T + p^{7} T^{2} \) |
| 89 | \( 1 - 9368136 T + p^{7} T^{2} \) |
| 97 | \( 1 + 2039995 T + p^{7} T^{2} \) |
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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
−9.979389782314883141918613288740, −9.370306183826006489759318995301, −8.092965395668492861779204548894, −7.23877751350999973218128064123, −6.09243213434735694097870436234, −5.39127460628073745962789552593, −4.31778926956106053724201203237, −2.70604875949189612528840640599, −2.16109793289111447436420634956, −0.61002085075839258380137267898,
0.61002085075839258380137267898, 2.16109793289111447436420634956, 2.70604875949189612528840640599, 4.31778926956106053724201203237, 5.39127460628073745962789552593, 6.09243213434735694097870436234, 7.23877751350999973218128064123, 8.092965395668492861779204548894, 9.370306183826006489759318995301, 9.979389782314883141918613288740
