| L(s) = 1 | − 3.20·3-s + 2.09·5-s − 1.61·7-s + 7.29·9-s − 3.47·11-s − 2.62·13-s − 6.72·15-s − 3.87·17-s − 4.93·19-s + 5.18·21-s − 2.23·23-s − 0.607·25-s − 13.7·27-s + 8.77·29-s + 3.42·31-s + 11.1·33-s − 3.38·35-s + 1.38·37-s + 8.42·39-s − 0.103·41-s + 7.46·43-s + 15.2·45-s − 10.4·47-s − 4.38·49-s + 12.4·51-s + 2.46·53-s − 7.28·55-s + ⋯ |
| L(s) = 1 | − 1.85·3-s + 0.937·5-s − 0.610·7-s + 2.43·9-s − 1.04·11-s − 0.728·13-s − 1.73·15-s − 0.938·17-s − 1.13·19-s + 1.13·21-s − 0.465·23-s − 0.121·25-s − 2.65·27-s + 1.62·29-s + 0.615·31-s + 1.94·33-s − 0.572·35-s + 0.227·37-s + 1.34·39-s − 0.0161·41-s + 1.13·43-s + 2.27·45-s − 1.52·47-s − 0.626·49-s + 1.73·51-s + 0.338·53-s − 0.982·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5292256824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5292256824\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 3.20T + 3T^{2} \) |
| 5 | \( 1 - 2.09T + 5T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 + 3.47T + 11T^{2} \) |
| 13 | \( 1 + 2.62T + 13T^{2} \) |
| 17 | \( 1 + 3.87T + 17T^{2} \) |
| 19 | \( 1 + 4.93T + 19T^{2} \) |
| 23 | \( 1 + 2.23T + 23T^{2} \) |
| 29 | \( 1 - 8.77T + 29T^{2} \) |
| 31 | \( 1 - 3.42T + 31T^{2} \) |
| 37 | \( 1 - 1.38T + 37T^{2} \) |
| 41 | \( 1 + 0.103T + 41T^{2} \) |
| 43 | \( 1 - 7.46T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 2.46T + 53T^{2} \) |
| 59 | \( 1 - 5.54T + 59T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 7.44T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 5.59T + 83T^{2} \) |
| 89 | \( 1 + 0.398T + 89T^{2} \) |
| 97 | \( 1 + 5.67T + 97T^{2} \) |
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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
−8.410824769927545366511066132634, −7.46155005230629568968124390921, −6.47561851773478709201690606280, −6.36996948226835665141823143738, −5.56284221207588507165412209762, −4.82298822806887637363841784769, −4.30896797107503769256176631506, −2.73770233166865509572165729764, −1.85208384051619209534092976633, −0.44015986935736294836688545063,
0.44015986935736294836688545063, 1.85208384051619209534092976633, 2.73770233166865509572165729764, 4.30896797107503769256176631506, 4.82298822806887637363841784769, 5.56284221207588507165412209762, 6.36996948226835665141823143738, 6.47561851773478709201690606280, 7.46155005230629568968124390921, 8.410824769927545366511066132634
