L(s) = 1 | + 2-s + 4-s − 3.69·5-s + 8-s − 3.69·10-s − 1.47·11-s − 2.69·13-s + 16-s + 6.57·17-s − 0.888·19-s − 3.69·20-s − 1.47·22-s + 6.28·23-s + 8.68·25-s − 2.69·26-s − 2.51·29-s − 6.81·31-s + 32-s + 6.57·34-s + 2.77·37-s − 0.888·38-s − 3.69·40-s − 4.11·41-s − 0.0123·43-s − 1.47·44-s + 6.28·46-s + 6.98·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.65·5-s + 0.353·8-s − 1.16·10-s − 0.445·11-s − 0.748·13-s + 0.250·16-s + 1.59·17-s − 0.203·19-s − 0.827·20-s − 0.314·22-s + 1.31·23-s + 1.73·25-s − 0.529·26-s − 0.466·29-s − 1.22·31-s + 0.176·32-s + 1.12·34-s + 0.456·37-s − 0.144·38-s − 0.584·40-s − 0.642·41-s − 0.00188·43-s − 0.222·44-s + 0.927·46-s + 1.01·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 3.69T + 5T^{2} \) |
| 11 | \( 1 + 1.47T + 11T^{2} \) |
| 13 | \( 1 + 2.69T + 13T^{2} \) |
| 17 | \( 1 - 6.57T + 17T^{2} \) |
| 19 | \( 1 + 0.888T + 19T^{2} \) |
| 23 | \( 1 - 6.28T + 23T^{2} \) |
| 29 | \( 1 + 2.51T + 29T^{2} \) |
| 31 | \( 1 + 6.81T + 31T^{2} \) |
| 37 | \( 1 - 2.77T + 37T^{2} \) |
| 41 | \( 1 + 4.11T + 41T^{2} \) |
| 43 | \( 1 + 0.0123T + 43T^{2} \) |
| 47 | \( 1 - 6.98T + 47T^{2} \) |
| 53 | \( 1 - 3.21T + 53T^{2} \) |
| 59 | \( 1 + 6.90T + 59T^{2} \) |
| 61 | \( 1 - 5.73T + 61T^{2} \) |
| 67 | \( 1 + 9.46T + 67T^{2} \) |
| 71 | \( 1 + 5.46T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 4.47T + 83T^{2} \) |
| 89 | \( 1 + 8.87T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−7.28917052619897682577679915330, −7.19606043006391944896064971058, −5.95292438892158823661206157958, −5.23072977470452052331606820807, −4.66832802371027117807892567809, −3.84199466312935727543853423258, −3.30366588790049453212321153100, −2.58685017450413322915969489030, −1.21366501683653714453657033846, 0,
1.21366501683653714453657033846, 2.58685017450413322915969489030, 3.30366588790049453212321153100, 3.84199466312935727543853423258, 4.66832802371027117807892567809, 5.23072977470452052331606820807, 5.95292438892158823661206157958, 7.19606043006391944896064971058, 7.28917052619897682577679915330