| L(s) = 1 | + 3-s + 5-s − 1.64·7-s + 9-s − 0.354·11-s − 0.354·13-s + 15-s − 4·17-s + 19-s − 1.64·21-s − 9.29·23-s + 25-s + 27-s + 8.93·29-s − 6·31-s − 0.354·33-s − 1.64·35-s + 3.64·37-s − 0.354·39-s − 9.64·41-s − 5.64·43-s + 45-s + 1.29·47-s − 4.29·49-s − 4·51-s + 11.2·53-s − 0.354·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.622·7-s + 0.333·9-s − 0.106·11-s − 0.0982·13-s + 0.258·15-s − 0.970·17-s + 0.229·19-s − 0.359·21-s − 1.93·23-s + 0.200·25-s + 0.192·27-s + 1.65·29-s − 1.07·31-s − 0.0616·33-s − 0.278·35-s + 0.599·37-s − 0.0567·39-s − 1.50·41-s − 0.860·43-s + 0.149·45-s + 0.188·47-s − 0.613·49-s − 0.560·51-s + 1.55·53-s − 0.0477·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 1.64T + 7T^{2} \) |
| 11 | \( 1 + 0.354T + 11T^{2} \) |
| 13 | \( 1 + 0.354T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 23 | \( 1 + 9.29T + 23T^{2} \) |
| 29 | \( 1 - 8.93T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 3.64T + 37T^{2} \) |
| 41 | \( 1 + 9.64T + 41T^{2} \) |
| 43 | \( 1 + 5.64T + 43T^{2} \) |
| 47 | \( 1 - 1.29T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 6.58T + 67T^{2} \) |
| 71 | \( 1 + 7.29T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 6.58T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 + 2.93T + 97T^{2} \) |
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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
−8.103202201267605249542061926087, −7.20804296654026188462751474235, −6.50936628341043363311237784174, −5.92757784137721078706151081570, −4.91843815644440497652452897532, −4.13254558365231013874430078474, −3.26443391341466734443330600764, −2.44701737521964160679937682072, −1.60777651339802500577588911250, 0,
1.60777651339802500577588911250, 2.44701737521964160679937682072, 3.26443391341466734443330600764, 4.13254558365231013874430078474, 4.91843815644440497652452897532, 5.92757784137721078706151081570, 6.50936628341043363311237784174, 7.20804296654026188462751474235, 8.103202201267605249542061926087
