L(s) = 1 | − 3.40·3-s + 1.59·5-s + 2.94·7-s + 8.62·9-s − 4.17·11-s + 2.42·13-s − 5.42·15-s + 17-s + 2.92·19-s − 10.0·21-s − 5.49·23-s − 2.46·25-s − 19.1·27-s + 6.57·29-s − 9.00·31-s + 14.2·33-s + 4.68·35-s − 9.65·37-s − 8.27·39-s − 6.88·41-s + 8.59·43-s + 13.7·45-s − 9.92·47-s + 1.67·49-s − 3.40·51-s + 7.14·53-s − 6.63·55-s + ⋯ |
L(s) = 1 | − 1.96·3-s + 0.711·5-s + 1.11·7-s + 2.87·9-s − 1.25·11-s + 0.673·13-s − 1.40·15-s + 0.242·17-s + 0.670·19-s − 2.19·21-s − 1.14·23-s − 0.493·25-s − 3.68·27-s + 1.22·29-s − 1.61·31-s + 2.47·33-s + 0.792·35-s − 1.58·37-s − 1.32·39-s − 1.07·41-s + 1.31·43-s + 2.04·45-s − 1.44·47-s + 0.239·49-s − 0.477·51-s + 0.982·53-s − 0.895·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 + 3.40T + 3T^{2} \) |
| 5 | \( 1 - 1.59T + 5T^{2} \) |
| 7 | \( 1 - 2.94T + 7T^{2} \) |
| 11 | \( 1 + 4.17T + 11T^{2} \) |
| 13 | \( 1 - 2.42T + 13T^{2} \) |
| 19 | \( 1 - 2.92T + 19T^{2} \) |
| 23 | \( 1 + 5.49T + 23T^{2} \) |
| 29 | \( 1 - 6.57T + 29T^{2} \) |
| 31 | \( 1 + 9.00T + 31T^{2} \) |
| 37 | \( 1 + 9.65T + 37T^{2} \) |
| 41 | \( 1 + 6.88T + 41T^{2} \) |
| 43 | \( 1 - 8.59T + 43T^{2} \) |
| 47 | \( 1 + 9.92T + 47T^{2} \) |
| 53 | \( 1 - 7.14T + 53T^{2} \) |
| 61 | \( 1 - 2.68T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 2.57T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 - 5.62T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 3.27T + 89T^{2} \) |
| 97 | \( 1 - 5.72T + 97T^{2} \) |
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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
−7.80726343786998312902509955926, −7.31917024212943245582576232782, −6.31795561090904505738182216258, −5.74033925271275853975835088745, −5.21504523944744937210038007555, −4.72737649941295110487915843913, −3.66347384709213451686146777643, −2.00945012594455727412844236683, −1.33644365673593722320694075076, 0,
1.33644365673593722320694075076, 2.00945012594455727412844236683, 3.66347384709213451686146777643, 4.72737649941295110487915843913, 5.21504523944744937210038007555, 5.74033925271275853975835088745, 6.31795561090904505738182216258, 7.31917024212943245582576232782, 7.80726343786998312902509955926