| L(s) = 1 | − 8·3-s − 16·7-s + 37·9-s − 40·11-s + 50·13-s + 30·17-s + 40·19-s + 128·21-s − 48·23-s − 80·27-s − 34·29-s + 320·31-s + 320·33-s − 310·37-s − 400·39-s + 410·41-s − 152·43-s + 416·47-s − 87·49-s − 240·51-s + 410·53-s − 320·57-s − 200·59-s + 30·61-s − 592·63-s − 776·67-s + 384·69-s + ⋯ |
| L(s) = 1 | − 1.53·3-s − 0.863·7-s + 1.37·9-s − 1.09·11-s + 1.06·13-s + 0.428·17-s + 0.482·19-s + 1.33·21-s − 0.435·23-s − 0.570·27-s − 0.217·29-s + 1.85·31-s + 1.68·33-s − 1.37·37-s − 1.64·39-s + 1.56·41-s − 0.539·43-s + 1.29·47-s − 0.253·49-s − 0.658·51-s + 1.06·53-s − 0.743·57-s − 0.441·59-s + 0.0629·61-s − 1.18·63-s − 1.41·67-s + 0.669·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 8 T + p^{3} T^{2} \) |
| 7 | \( 1 + 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 40 T + p^{3} T^{2} \) |
| 13 | \( 1 - 50 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 40 T + p^{3} T^{2} \) |
| 23 | \( 1 + 48 T + p^{3} T^{2} \) |
| 29 | \( 1 + 34 T + p^{3} T^{2} \) |
| 31 | \( 1 - 320 T + p^{3} T^{2} \) |
| 37 | \( 1 + 310 T + p^{3} T^{2} \) |
| 41 | \( 1 - 10 p T + p^{3} T^{2} \) |
| 43 | \( 1 + 152 T + p^{3} T^{2} \) |
| 47 | \( 1 - 416 T + p^{3} T^{2} \) |
| 53 | \( 1 - 410 T + p^{3} T^{2} \) |
| 59 | \( 1 + 200 T + p^{3} T^{2} \) |
| 61 | \( 1 - 30 T + p^{3} T^{2} \) |
| 67 | \( 1 + 776 T + p^{3} T^{2} \) |
| 71 | \( 1 - 400 T + p^{3} T^{2} \) |
| 73 | \( 1 - 630 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1120 T + p^{3} T^{2} \) |
| 83 | \( 1 + 552 T + p^{3} T^{2} \) |
| 89 | \( 1 + 326 T + p^{3} T^{2} \) |
| 97 | \( 1 - 110 T + p^{3} T^{2} \) |
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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
−9.810347723658777779352310817592, −8.594668388698051222446268819287, −7.54036907281691373524872914320, −6.56969777375511160861450165537, −5.88601219545492933940468517298, −5.24293049573164040244437465990, −4.09134926018868025829708198865, −2.85336927839034424867973599003, −1.08105490248722979296545288101, 0,
1.08105490248722979296545288101, 2.85336927839034424867973599003, 4.09134926018868025829708198865, 5.24293049573164040244437465990, 5.88601219545492933940468517298, 6.56969777375511160861450165537, 7.54036907281691373524872914320, 8.594668388698051222446268819287, 9.810347723658777779352310817592
