| L(s) = 1 | + 2.81·3-s + 4.18·5-s − 4.08·7-s + 4.92·9-s − 11-s + 2.68·13-s + 11.7·15-s − 2.52·17-s − 19-s − 11.5·21-s + 4.36·23-s + 12.5·25-s + 5.41·27-s − 4.29·29-s + 8.43·31-s − 2.81·33-s − 17.1·35-s + 7.26·37-s + 7.54·39-s + 7.41·41-s + 6.55·43-s + 20.6·45-s − 7.36·47-s + 9.70·49-s − 7.11·51-s − 9.47·53-s − 4.18·55-s + ⋯ |
| L(s) = 1 | + 1.62·3-s + 1.87·5-s − 1.54·7-s + 1.64·9-s − 0.301·11-s + 0.743·13-s + 3.04·15-s − 0.612·17-s − 0.229·19-s − 2.51·21-s + 0.909·23-s + 2.51·25-s + 1.04·27-s − 0.798·29-s + 1.51·31-s − 0.489·33-s − 2.89·35-s + 1.19·37-s + 1.20·39-s + 1.15·41-s + 0.999·43-s + 3.07·45-s − 1.07·47-s + 1.38·49-s − 0.996·51-s − 1.30·53-s − 0.564·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.291597427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.291597427\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.81T + 3T^{2} \) |
| 5 | \( 1 - 4.18T + 5T^{2} \) |
| 7 | \( 1 + 4.08T + 7T^{2} \) |
| 13 | \( 1 - 2.68T + 13T^{2} \) |
| 17 | \( 1 + 2.52T + 17T^{2} \) |
| 23 | \( 1 - 4.36T + 23T^{2} \) |
| 29 | \( 1 + 4.29T + 29T^{2} \) |
| 31 | \( 1 - 8.43T + 31T^{2} \) |
| 37 | \( 1 - 7.26T + 37T^{2} \) |
| 41 | \( 1 - 7.41T + 41T^{2} \) |
| 43 | \( 1 - 6.55T + 43T^{2} \) |
| 47 | \( 1 + 7.36T + 47T^{2} \) |
| 53 | \( 1 + 9.47T + 53T^{2} \) |
| 59 | \( 1 - 4.38T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 2.44T + 71T^{2} \) |
| 73 | \( 1 + 15.8T + 73T^{2} \) |
| 79 | \( 1 - 7.62T + 79T^{2} \) |
| 83 | \( 1 - 8.83T + 83T^{2} \) |
| 89 | \( 1 - 6.54T + 89T^{2} \) |
| 97 | \( 1 + 8.16T + 97T^{2} \) |
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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
−9.038247317406439323868904697831, −8.058264812469961156815080784335, −7.06635097099485527341680187452, −6.33143186418426744165529938207, −5.91352756577997876872170876334, −4.68037248337921222526285605661, −3.61189693602015652583701060110, −2.73289805680166479042187673309, −2.44696049089228524954824809622, −1.24745544838439043486173497037,
1.24745544838439043486173497037, 2.44696049089228524954824809622, 2.73289805680166479042187673309, 3.61189693602015652583701060110, 4.68037248337921222526285605661, 5.91352756577997876872170876334, 6.33143186418426744165529938207, 7.06635097099485527341680187452, 8.058264812469961156815080784335, 9.038247317406439323868904697831
