| L(s) = 1 | − 1.31·5-s + 1.31·7-s + 2.25·11-s − 0.348·13-s + 2·17-s − 19-s − 4.85·23-s − 3.27·25-s + 6.18·29-s + 4.20·31-s − 1.72·35-s − 9.13·37-s − 9.39·41-s − 4.85·43-s − 2.59·47-s − 5.27·49-s + 5.91·53-s − 2.96·55-s + 0.964·59-s − 4.93·61-s + 0.457·65-s − 2.38·67-s + 0.167·71-s + 12.9·73-s + 2.96·77-s − 7.09·79-s − 3.64·83-s + ⋯ |
| L(s) = 1 | − 0.587·5-s + 0.496·7-s + 0.680·11-s − 0.0966·13-s + 0.485·17-s − 0.229·19-s − 1.01·23-s − 0.655·25-s + 1.14·29-s + 0.755·31-s − 0.291·35-s − 1.50·37-s − 1.46·41-s − 0.740·43-s − 0.378·47-s − 0.753·49-s + 0.813·53-s − 0.399·55-s + 0.125·59-s − 0.632·61-s + 0.0567·65-s − 0.291·67-s + 0.0199·71-s + 1.51·73-s + 0.337·77-s − 0.798·79-s − 0.399·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 1.31T + 5T^{2} \) |
| 7 | \( 1 - 1.31T + 7T^{2} \) |
| 11 | \( 1 - 2.25T + 11T^{2} \) |
| 13 | \( 1 + 0.348T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 23 | \( 1 + 4.85T + 23T^{2} \) |
| 29 | \( 1 - 6.18T + 29T^{2} \) |
| 31 | \( 1 - 4.20T + 31T^{2} \) |
| 37 | \( 1 + 9.13T + 37T^{2} \) |
| 41 | \( 1 + 9.39T + 41T^{2} \) |
| 43 | \( 1 + 4.85T + 43T^{2} \) |
| 47 | \( 1 + 2.59T + 47T^{2} \) |
| 53 | \( 1 - 5.91T + 53T^{2} \) |
| 59 | \( 1 - 0.964T + 59T^{2} \) |
| 61 | \( 1 + 4.93T + 61T^{2} \) |
| 67 | \( 1 + 2.38T + 67T^{2} \) |
| 71 | \( 1 - 0.167T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 7.09T + 79T^{2} \) |
| 83 | \( 1 + 3.64T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 - 5.48T + 97T^{2} \) |
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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
−7.54863159750365154319350744590, −6.74487778177807167570488376578, −6.22951306103910292510220566177, −5.25543530134353022869410536872, −4.65389064387262796109528606277, −3.85243939737636757465060957035, −3.27271955170252057359544931452, −2.12047188617396541470571481922, −1.29319841479244047596214887018, 0,
1.29319841479244047596214887018, 2.12047188617396541470571481922, 3.27271955170252057359544931452, 3.85243939737636757465060957035, 4.65389064387262796109528606277, 5.25543530134353022869410536872, 6.22951306103910292510220566177, 6.74487778177807167570488376578, 7.54863159750365154319350744590
