| L(s) = 1 | + 2·2-s − 124·4-s + 125·5-s + 343·7-s − 504·8-s + 250·10-s − 2.72e3·11-s + 2.87e3·13-s + 686·14-s + 1.48e4·16-s − 9.27e3·17-s − 4.30e3·19-s − 1.55e4·20-s − 5.44e3·22-s + 4.15e4·23-s + 1.56e4·25-s + 5.74e3·26-s − 4.25e4·28-s + 3.54e4·29-s − 5.29e4·31-s + 9.42e4·32-s − 1.85e4·34-s + 4.28e4·35-s − 8.40e4·37-s − 8.60e3·38-s − 6.30e4·40-s − 1.80e5·41-s + ⋯ |
| L(s) = 1 | + 0.176·2-s − 0.968·4-s + 0.447·5-s + 0.377·7-s − 0.348·8-s + 0.0790·10-s − 0.617·11-s + 0.362·13-s + 0.0668·14-s + 0.907·16-s − 0.458·17-s − 0.143·19-s − 0.433·20-s − 0.109·22-s + 0.711·23-s + 1/5·25-s + 0.0641·26-s − 0.366·28-s + 0.270·29-s − 0.319·31-s + 0.508·32-s − 0.0809·34-s + 0.169·35-s − 0.272·37-s − 0.0254·38-s − 0.155·40-s − 0.408·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - p^{3} T \) |
| 7 | \( 1 - p^{3} T \) |
| good | 2 | \( 1 - p T + p^{7} T^{2} \) |
| 11 | \( 1 + 2724 T + p^{7} T^{2} \) |
| 13 | \( 1 - 2874 T + p^{7} T^{2} \) |
| 17 | \( 1 + 9278 T + p^{7} T^{2} \) |
| 19 | \( 1 + 4304 T + p^{7} T^{2} \) |
| 23 | \( 1 - 41500 T + p^{7} T^{2} \) |
| 29 | \( 1 - 35498 T + p^{7} T^{2} \) |
| 31 | \( 1 + 52940 T + p^{7} T^{2} \) |
| 37 | \( 1 + 84098 T + p^{7} T^{2} \) |
| 41 | \( 1 + 180342 T + p^{7} T^{2} \) |
| 43 | \( 1 + 33452 T + p^{7} T^{2} \) |
| 47 | \( 1 - 136120 T + p^{7} T^{2} \) |
| 53 | \( 1 - 23974 p T + p^{7} T^{2} \) |
| 59 | \( 1 - 1553252 T + p^{7} T^{2} \) |
| 61 | \( 1 - 213598 T + p^{7} T^{2} \) |
| 67 | \( 1 - 487228 T + p^{7} T^{2} \) |
| 71 | \( 1 + 1086000 T + p^{7} T^{2} \) |
| 73 | \( 1 + 5921978 T + p^{7} T^{2} \) |
| 79 | \( 1 + 5429824 T + p^{7} T^{2} \) |
| 83 | \( 1 + 6933404 T + p^{7} T^{2} \) |
| 89 | \( 1 + 262614 T + p^{7} T^{2} \) |
| 97 | \( 1 + 522234 T + p^{7} T^{2} \) |
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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
−9.978714357961983819834016362634, −8.941675942387986477726587270182, −8.324389533298515320035954047280, −7.08617885469246626824000488682, −5.78613140817393934315282691278, −5.01104748816947260210337486803, −4.00425285494690616056593847642, −2.71048883429078753857118637643, −1.28177727953048994887417205121, 0,
1.28177727953048994887417205121, 2.71048883429078753857118637643, 4.00425285494690616056593847642, 5.01104748816947260210337486803, 5.78613140817393934315282691278, 7.08617885469246626824000488682, 8.324389533298515320035954047280, 8.941675942387986477726587270182, 9.978714357961983819834016362634
