| L(s) = 1 | + 3-s − 3.05·5-s + 2.59·7-s + 9-s + 4.72·11-s + 6.74·13-s − 3.05·15-s − 3.13·17-s + 2.68·19-s + 2.59·21-s + 6.99·23-s + 4.33·25-s + 27-s − 7.69·29-s + 4.05·31-s + 4.72·33-s − 7.94·35-s + 0.127·37-s + 6.74·39-s − 8.00·41-s − 10.7·43-s − 3.05·45-s + 1.96·47-s − 0.240·49-s − 3.13·51-s − 7.56·53-s − 14.4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.36·5-s + 0.982·7-s + 0.333·9-s + 1.42·11-s + 1.86·13-s − 0.789·15-s − 0.760·17-s + 0.616·19-s + 0.567·21-s + 1.45·23-s + 0.867·25-s + 0.192·27-s − 1.42·29-s + 0.728·31-s + 0.821·33-s − 1.34·35-s + 0.0209·37-s + 1.07·39-s − 1.24·41-s − 1.63·43-s − 0.455·45-s + 0.286·47-s − 0.0343·49-s − 0.439·51-s − 1.03·53-s − 1.94·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.575911057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.575911057\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 3.05T + 5T^{2} \) |
| 7 | \( 1 - 2.59T + 7T^{2} \) |
| 11 | \( 1 - 4.72T + 11T^{2} \) |
| 13 | \( 1 - 6.74T + 13T^{2} \) |
| 17 | \( 1 + 3.13T + 17T^{2} \) |
| 19 | \( 1 - 2.68T + 19T^{2} \) |
| 23 | \( 1 - 6.99T + 23T^{2} \) |
| 29 | \( 1 + 7.69T + 29T^{2} \) |
| 31 | \( 1 - 4.05T + 31T^{2} \) |
| 37 | \( 1 - 0.127T + 37T^{2} \) |
| 41 | \( 1 + 8.00T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 1.96T + 47T^{2} \) |
| 53 | \( 1 + 7.56T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 2.45T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 + 3.69T + 71T^{2} \) |
| 73 | \( 1 - 4.93T + 73T^{2} \) |
| 79 | \( 1 - 9.15T + 79T^{2} \) |
| 83 | \( 1 - 7.82T + 83T^{2} \) |
| 89 | \( 1 - 5.37T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−8.377386501184480622806472676370, −7.981186737688373232205634808090, −6.95563666222168130712397559194, −6.57945924359235069648626745021, −5.28973270642469664416479254264, −4.45835340868453031491051957939, −3.70164906369068984996306503328, −3.36821685555089403684815479370, −1.78311566395289816103888612372, −0.986960748990390250358938163137,
0.986960748990390250358938163137, 1.78311566395289816103888612372, 3.36821685555089403684815479370, 3.70164906369068984996306503328, 4.45835340868453031491051957939, 5.28973270642469664416479254264, 6.57945924359235069648626745021, 6.95563666222168130712397559194, 7.981186737688373232205634808090, 8.377386501184480622806472676370
