| L(s) = 1 | + 2.02·2-s + 1.50·3-s + 2.11·4-s − 1.59·5-s + 3.06·6-s + 3.53·7-s + 0.229·8-s − 0.723·9-s − 3.23·10-s + 1.43·11-s + 3.18·12-s + 5.59·13-s + 7.17·14-s − 2.41·15-s − 3.76·16-s − 17-s − 1.46·18-s + 2.55·19-s − 3.37·20-s + 5.33·21-s + 2.90·22-s − 1.05·23-s + 0.346·24-s − 2.44·25-s + 11.3·26-s − 5.61·27-s + 7.47·28-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 0.871·3-s + 1.05·4-s − 0.714·5-s + 1.24·6-s + 1.33·7-s + 0.0811·8-s − 0.241·9-s − 1.02·10-s + 0.431·11-s + 0.920·12-s + 1.55·13-s + 1.91·14-s − 0.622·15-s − 0.940·16-s − 0.242·17-s − 0.345·18-s + 0.586·19-s − 0.754·20-s + 1.16·21-s + 0.618·22-s − 0.220·23-s + 0.0707·24-s − 0.489·25-s + 2.22·26-s − 1.08·27-s + 1.41·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.069636214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.069636214\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 2.02T + 2T^{2} \) |
| 3 | \( 1 - 1.50T + 3T^{2} \) |
| 5 | \( 1 + 1.59T + 5T^{2} \) |
| 7 | \( 1 - 3.53T + 7T^{2} \) |
| 11 | \( 1 - 1.43T + 11T^{2} \) |
| 13 | \( 1 - 5.59T + 13T^{2} \) |
| 19 | \( 1 - 2.55T + 19T^{2} \) |
| 23 | \( 1 + 1.05T + 23T^{2} \) |
| 29 | \( 1 - 1.56T + 29T^{2} \) |
| 31 | \( 1 - 3.98T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 4.17T + 41T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 - 8.88T + 59T^{2} \) |
| 61 | \( 1 + 4.99T + 61T^{2} \) |
| 67 | \( 1 - 9.04T + 67T^{2} \) |
| 71 | \( 1 - 0.338T + 71T^{2} \) |
| 73 | \( 1 + 8.43T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 + 7.23T + 83T^{2} \) |
| 89 | \( 1 - 7.05T + 89T^{2} \) |
| 97 | \( 1 + 1.19T + 97T^{2} \) |
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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
−10.85596696776966358749164238504, −9.299199370697523002069753211409, −8.391755373514375674039773993845, −7.989540117097471175101424690501, −6.71024121368494561037880770401, −5.69755009737934097297224544049, −4.72486435890019310826766340535, −3.84498860574206714756340775048, −3.19951914070826744362287497171, −1.77887449536406657185359192151,
1.77887449536406657185359192151, 3.19951914070826744362287497171, 3.84498860574206714756340775048, 4.72486435890019310826766340535, 5.69755009737934097297224544049, 6.71024121368494561037880770401, 7.989540117097471175101424690501, 8.391755373514375674039773993845, 9.299199370697523002069753211409, 10.85596696776966358749164238504
