| L(s) = 1 | + 8·2-s − 3·3-s + 64·4-s − 24·6-s + 343·7-s + 512·8-s − 2.17e3·9-s + 2.30e3·11-s − 192·12-s − 1.38e3·13-s + 2.74e3·14-s + 4.09e3·16-s + 4.00e3·17-s − 1.74e4·18-s − 7.68e3·19-s − 1.02e3·21-s + 1.84e4·22-s − 8.18e4·23-s − 1.53e3·24-s − 1.10e4·26-s + 1.30e4·27-s + 2.19e4·28-s + 1.57e5·29-s − 3.98e4·31-s + 3.27e4·32-s − 6.90e3·33-s + 3.20e4·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0641·3-s + 1/2·4-s − 0.0453·6-s + 0.377·7-s + 0.353·8-s − 0.995·9-s + 0.521·11-s − 0.0320·12-s − 0.174·13-s + 0.267·14-s + 1/4·16-s + 0.197·17-s − 0.704·18-s − 0.257·19-s − 0.0242·21-s + 0.368·22-s − 1.40·23-s − 0.0226·24-s − 0.123·26-s + 0.128·27-s + 0.188·28-s + 1.19·29-s − 0.240·31-s + 0.176·32-s − 0.0334·33-s + 0.139·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{3} T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p^{3} T \) |
| good | 3 | \( 1 + p T + p^{7} T^{2} \) |
| 11 | \( 1 - 2303 T + p^{7} T^{2} \) |
| 13 | \( 1 + 1381 T + p^{7} T^{2} \) |
| 17 | \( 1 - 4009 T + p^{7} T^{2} \) |
| 19 | \( 1 + 7688 T + p^{7} T^{2} \) |
| 23 | \( 1 + 81810 T + p^{7} T^{2} \) |
| 29 | \( 1 - 157191 T + p^{7} T^{2} \) |
| 31 | \( 1 + 39834 T + p^{7} T^{2} \) |
| 37 | \( 1 + 125266 T + p^{7} T^{2} \) |
| 41 | \( 1 + 739014 T + p^{7} T^{2} \) |
| 43 | \( 1 + 294604 T + p^{7} T^{2} \) |
| 47 | \( 1 - 655397 T + p^{7} T^{2} \) |
| 53 | \( 1 + 291934 T + p^{7} T^{2} \) |
| 59 | \( 1 + 2541922 T + p^{7} T^{2} \) |
| 61 | \( 1 - 1437280 T + p^{7} T^{2} \) |
| 67 | \( 1 + 3150966 T + p^{7} T^{2} \) |
| 71 | \( 1 - 2117576 T + p^{7} T^{2} \) |
| 73 | \( 1 + 552310 T + p^{7} T^{2} \) |
| 79 | \( 1 + 2334419 T + p^{7} T^{2} \) |
| 83 | \( 1 + 219508 T + p^{7} T^{2} \) |
| 89 | \( 1 + 3150280 T + p^{7} T^{2} \) |
| 97 | \( 1 + 12182135 T + p^{7} T^{2} \) |
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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
−10.00637645089409539949527219390, −8.744012941477605926708436302641, −7.954392981898992248869852368228, −6.72354712423938948490815201421, −5.86599478195856465174680781884, −4.91474030772634368716795219246, −3.83249775043893482914201912623, −2.72439655978774501948294046012, −1.55151976801426159385973894071, 0,
1.55151976801426159385973894071, 2.72439655978774501948294046012, 3.83249775043893482914201912623, 4.91474030772634368716795219246, 5.86599478195856465174680781884, 6.72354712423938948490815201421, 7.954392981898992248869852368228, 8.744012941477605926708436302641, 10.00637645089409539949527219390
