| L(s) = 1 | − 2-s + 4-s + 1.61·5-s + 7-s − 8-s − 1.61·10-s − 0.618·13-s − 14-s + 16-s + 1.76·17-s + 0.854·19-s + 1.61·20-s − 6.09·23-s − 2.38·25-s + 0.618·26-s + 28-s + 7.09·29-s − 8.85·31-s − 32-s − 1.76·34-s + 1.61·35-s − 9·37-s − 0.854·38-s − 1.61·40-s + 0.381·41-s + 1.85·43-s + 6.09·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.723·5-s + 0.377·7-s − 0.353·8-s − 0.511·10-s − 0.171·13-s − 0.267·14-s + 0.250·16-s + 0.427·17-s + 0.195·19-s + 0.361·20-s − 1.26·23-s − 0.476·25-s + 0.121·26-s + 0.188·28-s + 1.31·29-s − 1.59·31-s − 0.176·32-s − 0.302·34-s + 0.273·35-s − 1.47·37-s − 0.138·38-s − 0.255·40-s + 0.0596·41-s + 0.282·43-s + 0.897·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 1.61T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 13 | \( 1 + 0.618T + 13T^{2} \) |
| 17 | \( 1 - 1.76T + 17T^{2} \) |
| 19 | \( 1 - 0.854T + 19T^{2} \) |
| 23 | \( 1 + 6.09T + 23T^{2} \) |
| 29 | \( 1 - 7.09T + 29T^{2} \) |
| 31 | \( 1 + 8.85T + 31T^{2} \) |
| 37 | \( 1 + 9T + 37T^{2} \) |
| 41 | \( 1 - 0.381T + 41T^{2} \) |
| 43 | \( 1 - 1.85T + 43T^{2} \) |
| 47 | \( 1 + 1.38T + 47T^{2} \) |
| 53 | \( 1 + 2.61T + 53T^{2} \) |
| 59 | \( 1 - 0.763T + 59T^{2} \) |
| 61 | \( 1 + 7.32T + 61T^{2} \) |
| 67 | \( 1 - 6.94T + 67T^{2} \) |
| 71 | \( 1 + 7.47T + 71T^{2} \) |
| 73 | \( 1 + 5.76T + 73T^{2} \) |
| 79 | \( 1 - 3T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 5.76T + 89T^{2} \) |
| 97 | \( 1 + 6.70T + 97T^{2} \) |
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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
−7.72156904211891878393075516330, −7.08646807741629319078866507747, −6.24933661335558480679783572160, −5.67010600443568182546168695448, −4.94437877827900210505047109124, −3.92440075701218024381447966830, −3.00506557374121736044586346331, −2.01964108586829996482774674266, −1.43449290461762195129521966107, 0,
1.43449290461762195129521966107, 2.01964108586829996482774674266, 3.00506557374121736044586346331, 3.92440075701218024381447966830, 4.94437877827900210505047109124, 5.67010600443568182546168695448, 6.24933661335558480679783572160, 7.08646807741629319078866507747, 7.72156904211891878393075516330
