| L(s) = 1 | + 1.49·2-s + 0.0947·3-s + 0.236·4-s + 5-s + 0.141·6-s − 4.82·7-s − 2.63·8-s − 2.99·9-s + 1.49·10-s + 1.06·11-s + 0.0224·12-s − 7.21·14-s + 0.0947·15-s − 4.41·16-s + 3.55·17-s − 4.47·18-s − 5.73·19-s + 0.236·20-s − 0.457·21-s + 1.59·22-s − 7.08·23-s − 0.249·24-s + 25-s − 0.567·27-s − 1.14·28-s + 1.47·29-s + 0.141·30-s + ⋯ |
| L(s) = 1 | + 1.05·2-s + 0.0547·3-s + 0.118·4-s + 0.447·5-s + 0.0578·6-s − 1.82·7-s − 0.932·8-s − 0.997·9-s + 0.472·10-s + 0.322·11-s + 0.00647·12-s − 1.92·14-s + 0.0244·15-s − 1.10·16-s + 0.863·17-s − 1.05·18-s − 1.31·19-s + 0.0528·20-s − 0.0998·21-s + 0.340·22-s − 1.47·23-s − 0.0510·24-s + 0.200·25-s − 0.109·27-s − 0.215·28-s + 0.273·29-s + 0.0258·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.49T + 2T^{2} \) |
| 3 | \( 1 - 0.0947T + 3T^{2} \) |
| 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 - 1.06T + 11T^{2} \) |
| 17 | \( 1 - 3.55T + 17T^{2} \) |
| 19 | \( 1 + 5.73T + 19T^{2} \) |
| 23 | \( 1 + 7.08T + 23T^{2} \) |
| 29 | \( 1 - 1.47T + 29T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 - 0.0253T + 37T^{2} \) |
| 41 | \( 1 + 0.267T + 41T^{2} \) |
| 43 | \( 1 - 3.55T + 43T^{2} \) |
| 47 | \( 1 + 6.51T + 47T^{2} \) |
| 53 | \( 1 - 0.991T + 53T^{2} \) |
| 59 | \( 1 + 8.72T + 59T^{2} \) |
| 61 | \( 1 - 6.33T + 61T^{2} \) |
| 67 | \( 1 + 5.17T + 67T^{2} \) |
| 71 | \( 1 - 7.76T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 8.78T + 79T^{2} \) |
| 83 | \( 1 - 0.725T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 - 3.43T + 97T^{2} \) |
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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
−9.720683032097752798471344398681, −9.092823571455958067964134928708, −8.173908577908493699588816809229, −6.61963108717394761867681932520, −6.15105317063718634909101564028, −5.51363975902661479498843279276, −4.17080804147010918033010023938, −3.34980360597810232805357348550, −2.52593323597768609941329051788, 0,
2.52593323597768609941329051788, 3.34980360597810232805357348550, 4.17080804147010918033010023938, 5.51363975902661479498843279276, 6.15105317063718634909101564028, 6.61963108717394761867681932520, 8.173908577908493699588816809229, 9.092823571455958067964134928708, 9.720683032097752798471344398681
