| L(s) = 1 | − 3·3-s − 1.46·5-s + 8.39·7-s + 9·9-s + 34.7·11-s + 13·13-s + 4.39·15-s − 108.·17-s + 143.·19-s − 25.1·21-s + 128.·23-s − 122.·25-s − 27·27-s + 18.8·29-s + 78.5·31-s − 104.·33-s − 12.2·35-s + 327.·37-s − 39·39-s + 327.·41-s − 336.·43-s − 13.1·45-s − 99.2·47-s − 272.·49-s + 324.·51-s + 686.·53-s − 50.9·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.130·5-s + 0.453·7-s + 0.333·9-s + 0.953·11-s + 0.277·13-s + 0.0756·15-s − 1.54·17-s + 1.72·19-s − 0.261·21-s + 1.16·23-s − 0.982·25-s − 0.192·27-s + 0.120·29-s + 0.455·31-s − 0.550·33-s − 0.0593·35-s + 1.45·37-s − 0.160·39-s + 1.24·41-s − 1.19·43-s − 0.0436·45-s − 0.308·47-s − 0.794·49-s + 0.890·51-s + 1.77·53-s − 0.124·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.109811203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.109811203\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 + 1.46T + 125T^{2} \) |
| 7 | \( 1 - 8.39T + 343T^{2} \) |
| 11 | \( 1 - 34.7T + 1.33e3T^{2} \) |
| 17 | \( 1 + 108.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 143.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 128.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 18.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 78.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 327.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 327.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 336.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 99.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 686.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 242.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 644.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 871.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 100.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 604.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 741.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 501.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.56e3T + 9.12e5T^{2} \) |
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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
−8.639134782559222820705833742138, −7.70914663507645320890912779024, −6.97813181518322587517449558663, −6.29379436037329359840412513949, −5.44435872815901686370455578666, −4.60547752360649103893513416751, −3.92425760569416123010242257170, −2.78466172469384241970553352331, −1.55136222901196661982519524271, −0.70686582964195287517465519461,
0.70686582964195287517465519461, 1.55136222901196661982519524271, 2.78466172469384241970553352331, 3.92425760569416123010242257170, 4.60547752360649103893513416751, 5.44435872815901686370455578666, 6.29379436037329359840412513949, 6.97813181518322587517449558663, 7.70914663507645320890912779024, 8.639134782559222820705833742138
