| L(s) = 1 | − 0.712·2-s − 1.49·4-s − 2.77·7-s + 2.48·8-s − 4.26·11-s + 0.779·13-s + 1.98·14-s + 1.21·16-s + 1.90·17-s + 6.72·19-s + 3.04·22-s − 2.17·23-s − 0.555·26-s + 4.14·28-s − 29-s − 8.82·31-s − 5.83·32-s − 1.35·34-s + 1.48·37-s − 4.78·38-s + 7.71·41-s − 8.19·43-s + 6.36·44-s + 1.54·46-s + 5.19·47-s + 0.727·49-s − 1.16·52-s + ⋯ |
| L(s) = 1 | − 0.503·2-s − 0.746·4-s − 1.05·7-s + 0.879·8-s − 1.28·11-s + 0.216·13-s + 0.529·14-s + 0.302·16-s + 0.461·17-s + 1.54·19-s + 0.648·22-s − 0.452·23-s − 0.108·26-s + 0.783·28-s − 0.185·29-s − 1.58·31-s − 1.03·32-s − 0.232·34-s + 0.243·37-s − 0.776·38-s + 1.20·41-s − 1.24·43-s + 0.960·44-s + 0.228·46-s + 0.757·47-s + 0.103·49-s − 0.161·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 0.712T + 2T^{2} \) |
| 7 | \( 1 + 2.77T + 7T^{2} \) |
| 11 | \( 1 + 4.26T + 11T^{2} \) |
| 13 | \( 1 - 0.779T + 13T^{2} \) |
| 17 | \( 1 - 1.90T + 17T^{2} \) |
| 19 | \( 1 - 6.72T + 19T^{2} \) |
| 23 | \( 1 + 2.17T + 23T^{2} \) |
| 31 | \( 1 + 8.82T + 31T^{2} \) |
| 37 | \( 1 - 1.48T + 37T^{2} \) |
| 41 | \( 1 - 7.71T + 41T^{2} \) |
| 43 | \( 1 + 8.19T + 43T^{2} \) |
| 47 | \( 1 - 5.19T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 4.46T + 59T^{2} \) |
| 61 | \( 1 + 5.24T + 61T^{2} \) |
| 67 | \( 1 + 8.49T + 67T^{2} \) |
| 71 | \( 1 + 0.663T + 71T^{2} \) |
| 73 | \( 1 - 16.5T + 73T^{2} \) |
| 79 | \( 1 - 9.54T + 79T^{2} \) |
| 83 | \( 1 - 0.0123T + 83T^{2} \) |
| 89 | \( 1 - 5.46T + 89T^{2} \) |
| 97 | \( 1 + 0.952T + 97T^{2} \) |
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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
−7.51998321050715498695386334560, −7.39557779205627741289473115590, −6.15778430321575164439314904428, −5.46293907034781336450100306864, −4.96477622157978970111524557238, −3.80505567277911078955317864042, −3.32132806073536577773722155801, −2.28426918983887133514478014970, −0.982930349876388716950656143583, 0,
0.982930349876388716950656143583, 2.28426918983887133514478014970, 3.32132806073536577773722155801, 3.80505567277911078955317864042, 4.96477622157978970111524557238, 5.46293907034781336450100306864, 6.15778430321575164439314904428, 7.39557779205627741289473115590, 7.51998321050715498695386334560
