| L(s) = 1 | + 1.59e6·3-s + 4.36e9·5-s − 3.11e11·7-s + 2.54e12·9-s + 6.06e13·11-s + 6.46e14·13-s + 6.96e15·15-s + 2.78e16·17-s + 3.23e17·19-s − 4.96e17·21-s − 2.77e18·23-s + 1.16e19·25-s + 4.05e18·27-s − 4.40e18·29-s − 1.35e20·31-s + 9.66e19·33-s − 1.35e21·35-s + 2.42e21·37-s + 1.03e21·39-s − 1.00e22·41-s − 6.87e21·43-s + 1.11e22·45-s + 7.32e22·47-s + 3.11e22·49-s + 4.44e22·51-s + 2.25e23·53-s + 2.64e23·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.60·5-s − 1.21·7-s + 0.333·9-s + 0.529·11-s + 0.592·13-s + 0.924·15-s + 0.681·17-s + 1.76·19-s − 0.700·21-s − 1.14·23-s + 1.56·25-s + 0.192·27-s − 0.0797·29-s − 0.995·31-s + 0.305·33-s − 1.94·35-s + 1.63·37-s + 0.341·39-s − 1.70·41-s − 0.610·43-s + 0.533·45-s + 1.95·47-s + 0.473·49-s + 0.393·51-s + 1.18·53-s + 0.847·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(14)\) |
\(\approx\) |
\(4.367831743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.367831743\) |
| \(L(\frac{29}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.59e6T \) |
| good | 5 | \( 1 - 4.36e9T + 7.45e18T^{2} \) |
| 7 | \( 1 + 3.11e11T + 6.57e22T^{2} \) |
| 11 | \( 1 - 6.06e13T + 1.31e28T^{2} \) |
| 13 | \( 1 - 6.46e14T + 1.19e30T^{2} \) |
| 17 | \( 1 - 2.78e16T + 1.66e33T^{2} \) |
| 19 | \( 1 - 3.23e17T + 3.36e34T^{2} \) |
| 23 | \( 1 + 2.77e18T + 5.84e36T^{2} \) |
| 29 | \( 1 + 4.40e18T + 3.05e39T^{2} \) |
| 31 | \( 1 + 1.35e20T + 1.84e40T^{2} \) |
| 37 | \( 1 - 2.42e21T + 2.19e42T^{2} \) |
| 41 | \( 1 + 1.00e22T + 3.50e43T^{2} \) |
| 43 | \( 1 + 6.87e21T + 1.26e44T^{2} \) |
| 47 | \( 1 - 7.32e22T + 1.40e45T^{2} \) |
| 53 | \( 1 - 2.25e23T + 3.59e46T^{2} \) |
| 59 | \( 1 - 6.81e22T + 6.50e47T^{2} \) |
| 61 | \( 1 - 2.06e24T + 1.59e48T^{2} \) |
| 67 | \( 1 + 5.18e24T + 2.01e49T^{2} \) |
| 71 | \( 1 + 9.12e24T + 9.63e49T^{2} \) |
| 73 | \( 1 + 6.99e24T + 2.04e50T^{2} \) |
| 79 | \( 1 + 3.46e25T + 1.72e51T^{2} \) |
| 83 | \( 1 - 5.24e25T + 6.53e51T^{2} \) |
| 89 | \( 1 - 2.05e26T + 4.30e52T^{2} \) |
| 97 | \( 1 - 3.43e26T + 4.39e53T^{2} \) |
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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
−10.16615095017602300676912623779, −9.690470117714450293190299982649, −8.865676492525466583574453318746, −7.30796439538500621827271816730, −6.18841066958626046258365738347, −5.50476369418244116681712293780, −3.75246637682478132245336427694, −2.88897734265127925217414309474, −1.80636949032861795819643728766, −0.865440736491405725008687668011,
0.865440736491405725008687668011, 1.80636949032861795819643728766, 2.88897734265127925217414309474, 3.75246637682478132245336427694, 5.50476369418244116681712293780, 6.18841066958626046258365738347, 7.30796439538500621827271816730, 8.865676492525466583574453318746, 9.690470117714450293190299982649, 10.16615095017602300676912623779
