| L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s − 230·5-s + 216·6-s + 106·7-s + 512·8-s + 729·9-s − 1.84e3·10-s − 4.32e3·11-s + 1.72e3·12-s − 5.42e3·13-s + 848·14-s − 6.21e3·15-s + 4.09e3·16-s − 3.30e3·17-s + 5.83e3·18-s − 4.14e3·19-s − 1.47e4·20-s + 2.86e3·21-s − 3.46e4·22-s + 1.21e4·23-s + 1.38e4·24-s − 2.52e4·25-s − 4.34e4·26-s + 1.96e4·27-s + 6.78e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.822·5-s + 0.408·6-s + 0.116·7-s + 0.353·8-s + 1/3·9-s − 0.581·10-s − 0.979·11-s + 0.288·12-s − 0.684·13-s + 0.0825·14-s − 0.475·15-s + 1/4·16-s − 0.162·17-s + 0.235·18-s − 0.138·19-s − 0.411·20-s + 0.0674·21-s − 0.692·22-s + 0.208·23-s + 0.204·24-s − 0.322·25-s − 0.484·26-s + 0.192·27-s + 0.0584·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - p^{3} T \) |
| 23 | \( 1 - p^{3} T \) |
| good | 5 | \( 1 + 46 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 106 T + p^{7} T^{2} \) |
| 11 | \( 1 + 4326 T + p^{7} T^{2} \) |
| 13 | \( 1 + 5426 T + p^{7} T^{2} \) |
| 17 | \( 1 + 3300 T + p^{7} T^{2} \) |
| 19 | \( 1 + 4140 T + p^{7} T^{2} \) |
| 29 | \( 1 + 150186 T + p^{7} T^{2} \) |
| 31 | \( 1 + 307192 T + p^{7} T^{2} \) |
| 37 | \( 1 + 55200 T + p^{7} T^{2} \) |
| 41 | \( 1 - 270130 T + p^{7} T^{2} \) |
| 43 | \( 1 - 36264 T + p^{7} T^{2} \) |
| 47 | \( 1 - 494224 T + p^{7} T^{2} \) |
| 53 | \( 1 - 646646 T + p^{7} T^{2} \) |
| 59 | \( 1 + 387948 T + p^{7} T^{2} \) |
| 61 | \( 1 + 2060876 T + p^{7} T^{2} \) |
| 67 | \( 1 + 17664 T + p^{7} T^{2} \) |
| 71 | \( 1 + 3580320 T + p^{7} T^{2} \) |
| 73 | \( 1 - 484550 T + p^{7} T^{2} \) |
| 79 | \( 1 + 2167314 T + p^{7} T^{2} \) |
| 83 | \( 1 - 381182 T + p^{7} T^{2} \) |
| 89 | \( 1 + 628620 T + p^{7} T^{2} \) |
| 97 | \( 1 - 13964418 T + p^{7} T^{2} \) |
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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
−11.45387726932621203135578480893, −10.51100450143727113001146019699, −9.170553799491089539159513148054, −7.83734111934987760741557825417, −7.25659446465695734983016659476, −5.57703695403937810358993020258, −4.39055811150266050200070308740, −3.28920176020151571282776256457, −2.04495799628407143847798118345, 0,
2.04495799628407143847798118345, 3.28920176020151571282776256457, 4.39055811150266050200070308740, 5.57703695403937810358993020258, 7.25659446465695734983016659476, 7.83734111934987760741557825417, 9.170553799491089539159513148054, 10.51100450143727113001146019699, 11.45387726932621203135578480893
