| L(s) = 1 | − 8·2-s + 64·4-s + 312·5-s + 323·7-s − 512·8-s − 2.49e3·10-s + 3.72e3·11-s − 1.41e4·13-s − 2.58e3·14-s + 4.09e3·16-s + 1.59e4·17-s + 2.24e4·19-s + 1.99e4·20-s − 2.97e4·22-s + 5.77e4·23-s + 1.92e4·25-s + 1.13e5·26-s + 2.06e4·28-s + 1.66e5·29-s + 9.48e4·31-s − 3.27e4·32-s − 1.27e5·34-s + 1.00e5·35-s + 4.53e5·37-s − 1.79e5·38-s − 1.59e5·40-s + 6.27e5·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.11·5-s + 0.355·7-s − 0.353·8-s − 0.789·10-s + 0.842·11-s − 1.78·13-s − 0.251·14-s + 1/4·16-s + 0.785·17-s + 0.749·19-s + 0.558·20-s − 0.595·22-s + 0.990·23-s + 0.246·25-s + 1.26·26-s + 0.177·28-s + 1.26·29-s + 0.571·31-s − 0.176·32-s − 0.555·34-s + 0.397·35-s + 1.47·37-s − 0.530·38-s − 0.394·40-s + 1.42·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.737425998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.737425998\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{3} T \) |
| 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
−14.10986249448926384184215775222, −12.57872526795046954890666865562, −11.45874053858735005378827256791, −9.933108613302544495155214691673, −9.437439314332290865020164651485, −7.82749439745330220182955636384, −6.51369962944040235986917280647, −5.04506462924663147351022830667, −2.62306784807073802752186041828, −1.13735543089555566756347930892,
1.13735543089555566756347930892, 2.62306784807073802752186041828, 5.04506462924663147351022830667, 6.51369962944040235986917280647, 7.82749439745330220182955636384, 9.437439314332290865020164651485, 9.933108613302544495155214691673, 11.45874053858735005378827256791, 12.57872526795046954890666865562, 14.10986249448926384184215775222
