| L(s) = 1 | − 2.96·3-s + 4.34·5-s + 5.77·9-s − 0.777·11-s − 0.473·13-s − 12.8·15-s + 1.81·17-s + 6.26·19-s + 5.33·23-s + 13.8·25-s − 8.21·27-s + 4.40·29-s − 5.82·31-s + 2.30·33-s − 8.34·37-s + 1.40·39-s + 41-s − 6.18·43-s + 25.0·45-s + 6.01·47-s − 5.36·51-s + 11.5·53-s − 3.37·55-s − 18.5·57-s − 0.805·59-s + 11.1·61-s − 2.05·65-s + ⋯ |
| L(s) = 1 | − 1.70·3-s + 1.94·5-s + 1.92·9-s − 0.234·11-s − 0.131·13-s − 3.32·15-s + 0.439·17-s + 1.43·19-s + 1.11·23-s + 2.77·25-s − 1.58·27-s + 0.818·29-s − 1.04·31-s + 0.400·33-s − 1.37·37-s + 0.224·39-s + 0.156·41-s − 0.942·43-s + 3.73·45-s + 0.877·47-s − 0.751·51-s + 1.59·53-s − 0.455·55-s − 2.45·57-s − 0.104·59-s + 1.42·61-s − 0.255·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.922027202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.922027202\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 2.96T + 3T^{2} \) |
| 5 | \( 1 - 4.34T + 5T^{2} \) |
| 11 | \( 1 + 0.777T + 11T^{2} \) |
| 13 | \( 1 + 0.473T + 13T^{2} \) |
| 17 | \( 1 - 1.81T + 17T^{2} \) |
| 19 | \( 1 - 6.26T + 19T^{2} \) |
| 23 | \( 1 - 5.33T + 23T^{2} \) |
| 29 | \( 1 - 4.40T + 29T^{2} \) |
| 31 | \( 1 + 5.82T + 31T^{2} \) |
| 37 | \( 1 + 8.34T + 37T^{2} \) |
| 43 | \( 1 + 6.18T + 43T^{2} \) |
| 47 | \( 1 - 6.01T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 0.805T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 1.94T + 71T^{2} \) |
| 73 | \( 1 - 0.839T + 73T^{2} \) |
| 79 | \( 1 - 7.38T + 79T^{2} \) |
| 83 | \( 1 - 9.38T + 83T^{2} \) |
| 89 | \( 1 + 1.70T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−7.40974964593754864080874391471, −6.93992095772863534796156403210, −6.28554123173001889433390668373, −5.64172817742191527206267184405, −5.19057374556804852661555678075, −4.90690369938854899586375063992, −3.49970762432093334516101517118, −2.47637288781717771262369656558, −1.48255044269548766941303595874, −0.828744725805483586060587084912,
0.828744725805483586060587084912, 1.48255044269548766941303595874, 2.47637288781717771262369656558, 3.49970762432093334516101517118, 4.90690369938854899586375063992, 5.19057374556804852661555678075, 5.64172817742191527206267184405, 6.28554123173001889433390668373, 6.93992095772863534796156403210, 7.40974964593754864080874391471
