| L(s) = 1 | − 5-s − 2.60·7-s − 11-s + 2.60·13-s + 0.605·17-s + 7.21·19-s + 25-s − 8·29-s − 5.21·31-s + 2.60·35-s + 11.2·37-s − 8·41-s − 10.6·43-s − 9.21·47-s − 0.211·49-s − 2·53-s + 55-s − 8·59-s − 7.21·61-s − 2.60·65-s − 4·67-s − 14.4·71-s − 6.60·73-s + 2.60·77-s + 3.21·79-s + 3.39·83-s − 0.605·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.984·7-s − 0.301·11-s + 0.722·13-s + 0.146·17-s + 1.65·19-s + 0.200·25-s − 1.48·29-s − 0.935·31-s + 0.440·35-s + 1.84·37-s − 1.24·41-s − 1.61·43-s − 1.34·47-s − 0.0301·49-s − 0.274·53-s + 0.134·55-s − 1.04·59-s − 0.923·61-s − 0.323·65-s − 0.488·67-s − 1.71·71-s − 0.773·73-s + 0.296·77-s + 0.361·79-s + 0.372·83-s − 0.0656·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 2.60T + 7T^{2} \) |
| 13 | \( 1 - 2.60T + 13T^{2} \) |
| 17 | \( 1 - 0.605T + 17T^{2} \) |
| 19 | \( 1 - 7.21T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 5.21T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + 9.21T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 7.21T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + 6.60T + 73T^{2} \) |
| 79 | \( 1 - 3.21T + 79T^{2} \) |
| 83 | \( 1 - 3.39T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−8.871310940855986168605964461869, −7.86516272920974114462482363779, −7.32646044243963158414398303659, −6.37428011716931618995064540840, −5.65626487790784522341557399974, −4.70771147374803911770920981402, −3.48267231180785381720246864589, −3.14549888582286074414369376069, −1.53599537243246885844567075786, 0,
1.53599537243246885844567075786, 3.14549888582286074414369376069, 3.48267231180785381720246864589, 4.70771147374803911770920981402, 5.65626487790784522341557399974, 6.37428011716931618995064540840, 7.32646044243963158414398303659, 7.86516272920974114462482363779, 8.871310940855986168605964461869
