| L(s) = 1 | + 2-s + 0.381·3-s + 4-s + 3.23·5-s + 0.381·6-s − 2·7-s + 8-s − 2.85·9-s + 3.23·10-s + 0.381·12-s + 1.23·13-s − 2·14-s + 1.23·15-s + 16-s − 0.618·17-s − 2.85·18-s − 5.85·19-s + 3.23·20-s − 0.763·21-s + 1.23·23-s + 0.381·24-s + 5.47·25-s + 1.23·26-s − 2.23·27-s − 2·28-s + 4.47·29-s + 1.23·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.220·3-s + 0.5·4-s + 1.44·5-s + 0.155·6-s − 0.755·7-s + 0.353·8-s − 0.951·9-s + 1.02·10-s + 0.110·12-s + 0.342·13-s − 0.534·14-s + 0.319·15-s + 0.250·16-s − 0.149·17-s − 0.672·18-s − 1.34·19-s + 0.723·20-s − 0.166·21-s + 0.257·23-s + 0.0779·24-s + 1.09·25-s + 0.242·26-s − 0.430·27-s − 0.377·28-s + 0.830·29-s + 0.225·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.110881752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.110881752\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 0.381T + 3T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 0.618T + 17T^{2} \) |
| 19 | \( 1 + 5.85T + 19T^{2} \) |
| 23 | \( 1 - 1.23T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 3.70T + 37T^{2} \) |
| 41 | \( 1 - 5.61T + 41T^{2} \) |
| 43 | \( 1 + 8.56T + 43T^{2} \) |
| 47 | \( 1 + 6.47T + 47T^{2} \) |
| 53 | \( 1 + 1.52T + 53T^{2} \) |
| 59 | \( 1 + 8.61T + 59T^{2} \) |
| 61 | \( 1 + 2.47T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 5.23T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 9.32T + 83T^{2} \) |
| 89 | \( 1 + 8.09T + 89T^{2} \) |
| 97 | \( 1 - 7.14T + 97T^{2} \) |
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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
−12.41591394572359680257222272584, −11.14308816516032217922027637895, −10.23988494776451817843304961024, −9.296910195920915486863152315656, −8.312168975313695718293713980838, −6.54097981734495754034534956685, −6.12040232995040826334216587585, −4.94563136271973363431212875724, −3.28338095505031098489531796514, −2.15746909252528951425940770847,
2.15746909252528951425940770847, 3.28338095505031098489531796514, 4.94563136271973363431212875724, 6.12040232995040826334216587585, 6.54097981734495754034534956685, 8.312168975313695718293713980838, 9.296910195920915486863152315656, 10.23988494776451817843304961024, 11.14308816516032217922027637895, 12.41591394572359680257222272584
