| L(s) = 1 | − 2.55·2-s + 4.52·4-s + 2.79·5-s − 0.153·7-s − 6.43·8-s − 7.14·10-s + 3.71·11-s − 1.28·13-s + 0.393·14-s + 7.39·16-s + 0.563·17-s − 6.62·19-s + 12.6·20-s − 9.48·22-s + 6.04·23-s + 2.83·25-s + 3.27·26-s − 0.696·28-s − 8.68·29-s − 1.69·31-s − 6.00·32-s − 1.43·34-s − 0.431·35-s + 4.07·37-s + 16.9·38-s − 18.0·40-s + 11.1·41-s + ⋯ |
| L(s) = 1 | − 1.80·2-s + 2.26·4-s + 1.25·5-s − 0.0581·7-s − 2.27·8-s − 2.26·10-s + 1.12·11-s − 0.356·13-s + 0.105·14-s + 1.84·16-s + 0.136·17-s − 1.51·19-s + 2.82·20-s − 2.02·22-s + 1.26·23-s + 0.567·25-s + 0.643·26-s − 0.131·28-s − 1.61·29-s − 0.304·31-s − 1.06·32-s − 0.246·34-s − 0.0728·35-s + 0.670·37-s + 2.74·38-s − 2.84·40-s + 1.73·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.036999748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.036999748\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 5 | \( 1 - 2.79T + 5T^{2} \) |
| 7 | \( 1 + 0.153T + 7T^{2} \) |
| 11 | \( 1 - 3.71T + 11T^{2} \) |
| 13 | \( 1 + 1.28T + 13T^{2} \) |
| 17 | \( 1 - 0.563T + 17T^{2} \) |
| 19 | \( 1 + 6.62T + 19T^{2} \) |
| 23 | \( 1 - 6.04T + 23T^{2} \) |
| 29 | \( 1 + 8.68T + 29T^{2} \) |
| 31 | \( 1 + 1.69T + 31T^{2} \) |
| 37 | \( 1 - 4.07T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 47 | \( 1 + 7.38T + 47T^{2} \) |
| 53 | \( 1 - 2.30T + 53T^{2} \) |
| 59 | \( 1 - 5.89T + 59T^{2} \) |
| 61 | \( 1 + 4.15T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 - 1.47T + 73T^{2} \) |
| 79 | \( 1 - 8.32T + 79T^{2} \) |
| 83 | \( 1 - 6.97T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−8.918179091246546405242045107966, −7.982806567932303798487665470678, −7.24014696208711544904910925651, −6.41810303097357974695457805252, −6.12742723550572270099493672808, −4.96915603263195286866470725991, −3.64515800564647230907539367249, −2.34935095279592412263039241824, −1.86224378092552091054156483468, −0.798863253716878346078528161638,
0.798863253716878346078528161638, 1.86224378092552091054156483468, 2.34935095279592412263039241824, 3.64515800564647230907539367249, 4.96915603263195286866470725991, 6.12742723550572270099493672808, 6.41810303097357974695457805252, 7.24014696208711544904910925651, 7.982806567932303798487665470678, 8.918179091246546405242045107966
