| L(s) = 1 | − 2.41·3-s + 0.585·5-s + 2.82·9-s − 11-s + 3.82·13-s − 1.41·15-s − 3.65·17-s − 0.585·19-s + 6.24·23-s − 4.65·25-s + 0.414·27-s + 2.65·29-s − 4·31-s + 2.41·33-s − 9.41·37-s − 9.24·39-s + 5.41·41-s + 5.65·43-s + 1.65·45-s − 10.4·47-s + 8.82·51-s + 7.89·53-s − 0.585·55-s + 1.41·57-s − 5.58·59-s − 11.8·61-s + 2.24·65-s + ⋯ |
| L(s) = 1 | − 1.39·3-s + 0.261·5-s + 0.942·9-s − 0.301·11-s + 1.06·13-s − 0.365·15-s − 0.886·17-s − 0.134·19-s + 1.30·23-s − 0.931·25-s + 0.0797·27-s + 0.493·29-s − 0.718·31-s + 0.420·33-s − 1.54·37-s − 1.48·39-s + 0.845·41-s + 0.862·43-s + 0.246·45-s − 1.52·47-s + 1.23·51-s + 1.08·53-s − 0.0789·55-s + 0.187·57-s − 0.727·59-s − 1.51·61-s + 0.278·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 - 0.585T + 5T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 + 0.585T + 19T^{2} \) |
| 23 | \( 1 - 6.24T + 23T^{2} \) |
| 29 | \( 1 - 2.65T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 9.41T + 37T^{2} \) |
| 41 | \( 1 - 5.41T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 7.89T + 53T^{2} \) |
| 59 | \( 1 + 5.58T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 2.75T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 - 9.41T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 3.82T + 97T^{2} \) |
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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
−7.18381533917848241124536941886, −6.57306077082138099561695325923, −6.06117767955046763830174684794, −5.40395469798647352576474926215, −4.84134828847548833819641991435, −4.04000089664478531465342148028, −3.13273606910473124857729844412, −2.01114596902434078080929526162, −1.06167590159195657203928361547, 0,
1.06167590159195657203928361547, 2.01114596902434078080929526162, 3.13273606910473124857729844412, 4.04000089664478531465342148028, 4.84134828847548833819641991435, 5.40395469798647352576474926215, 6.06117767955046763830174684794, 6.57306077082138099561695325923, 7.18381533917848241124536941886
