| L(s) = 1 | − 1.76·5-s + 2.40·7-s − 3·9-s + 4.64·11-s + 3.69·13-s + 1.00·17-s + 19-s − 2.66·23-s − 1.88·25-s − 0.522·29-s + 7.54·31-s − 4.24·35-s − 4.36·37-s + 9.51·41-s − 0.856·43-s + 5.29·45-s + 9.01·47-s − 1.22·49-s − 12.1·53-s − 8.19·55-s + 12.6·59-s − 2.58·61-s − 7.20·63-s − 6.53·65-s + 6.77·67-s − 2.48·71-s − 5.73·73-s + ⋯ |
| L(s) = 1 | − 0.789·5-s + 0.908·7-s − 9-s + 1.39·11-s + 1.02·13-s + 0.244·17-s + 0.229·19-s − 0.556·23-s − 0.376·25-s − 0.0970·29-s + 1.35·31-s − 0.717·35-s − 0.718·37-s + 1.48·41-s − 0.130·43-s + 0.789·45-s + 1.31·47-s − 0.175·49-s − 1.67·53-s − 1.10·55-s + 1.65·59-s − 0.331·61-s − 0.908·63-s − 0.810·65-s + 0.827·67-s − 0.295·71-s − 0.671·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.013349092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.013349092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 1.76T + 5T^{2} \) |
| 7 | \( 1 - 2.40T + 7T^{2} \) |
| 11 | \( 1 - 4.64T + 11T^{2} \) |
| 13 | \( 1 - 3.69T + 13T^{2} \) |
| 17 | \( 1 - 1.00T + 17T^{2} \) |
| 23 | \( 1 + 2.66T + 23T^{2} \) |
| 29 | \( 1 + 0.522T + 29T^{2} \) |
| 31 | \( 1 - 7.54T + 31T^{2} \) |
| 37 | \( 1 + 4.36T + 37T^{2} \) |
| 41 | \( 1 - 9.51T + 41T^{2} \) |
| 43 | \( 1 + 0.856T + 43T^{2} \) |
| 47 | \( 1 - 9.01T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 + 2.58T + 61T^{2} \) |
| 67 | \( 1 - 6.77T + 67T^{2} \) |
| 71 | \( 1 + 2.48T + 71T^{2} \) |
| 73 | \( 1 + 5.73T + 73T^{2} \) |
| 83 | \( 1 - 5.22T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 4.55T + 97T^{2} \) |
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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
−8.113385239934177939313633015365, −7.57918324014519034377528712404, −6.56371078893826405370555243789, −6.01785058799246263467425604222, −5.22540548258154970001720586413, −4.22133528727801328248160596674, −3.82624245465945797212437259714, −2.89363710043659577501309702926, −1.71708694913121798316435329284, −0.78437141148040842709414059503,
0.78437141148040842709414059503, 1.71708694913121798316435329284, 2.89363710043659577501309702926, 3.82624245465945797212437259714, 4.22133528727801328248160596674, 5.22540548258154970001720586413, 6.01785058799246263467425604222, 6.56371078893826405370555243789, 7.57918324014519034377528712404, 8.113385239934177939313633015365
