| L(s) = 1 | − 2.20·2-s − 2.49·3-s + 2.86·4-s + 5.51·6-s + 7-s − 1.91·8-s + 3.24·9-s − 2.20·11-s − 7.16·12-s − 5.51·13-s − 2.20·14-s − 1.50·16-s + 1.46·17-s − 7.15·18-s + 2.58·19-s − 2.49·21-s + 4.85·22-s + 8.52·23-s + 4.78·24-s + 12.1·26-s − 0.611·27-s + 2.86·28-s + 8.01·29-s − 5.09·31-s + 7.16·32-s + 5.49·33-s − 3.22·34-s + ⋯ |
| L(s) = 1 | − 1.56·2-s − 1.44·3-s + 1.43·4-s + 2.25·6-s + 0.377·7-s − 0.677·8-s + 1.08·9-s − 0.663·11-s − 2.06·12-s − 1.52·13-s − 0.589·14-s − 0.377·16-s + 0.354·17-s − 1.68·18-s + 0.593·19-s − 0.545·21-s + 1.03·22-s + 1.77·23-s + 0.977·24-s + 2.38·26-s − 0.117·27-s + 0.542·28-s + 1.48·29-s − 0.915·31-s + 1.26·32-s + 0.957·33-s − 0.552·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + 2.20T + 2T^{2} \) |
| 3 | \( 1 + 2.49T + 3T^{2} \) |
| 11 | \( 1 + 2.20T + 11T^{2} \) |
| 13 | \( 1 + 5.51T + 13T^{2} \) |
| 17 | \( 1 - 1.46T + 17T^{2} \) |
| 19 | \( 1 - 2.58T + 19T^{2} \) |
| 23 | \( 1 - 8.52T + 23T^{2} \) |
| 29 | \( 1 - 8.01T + 29T^{2} \) |
| 31 | \( 1 + 5.09T + 31T^{2} \) |
| 37 | \( 1 + 1.35T + 37T^{2} \) |
| 41 | \( 1 + 3.94T + 41T^{2} \) |
| 43 | \( 1 + 8.29T + 43T^{2} \) |
| 47 | \( 1 - 8.60T + 47T^{2} \) |
| 53 | \( 1 + 4.88T + 53T^{2} \) |
| 59 | \( 1 + 14.7T + 59T^{2} \) |
| 61 | \( 1 - 2.64T + 61T^{2} \) |
| 67 | \( 1 - 8.63T + 67T^{2} \) |
| 71 | \( 1 - 0.518T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 4.53T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 7.74T + 89T^{2} \) |
| 97 | \( 1 + 9.25T + 97T^{2} \) |
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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
−9.857923215557343774523713570080, −9.042786150763052742751229253250, −7.979301539160255844409173997921, −7.24787274107521824847853311187, −6.60956909567927065497434824284, −5.24795006489362574006876923437, −4.84723158480412154608599408122, −2.73209142148151271547547781254, −1.23083080202159175986424242816, 0,
1.23083080202159175986424242816, 2.73209142148151271547547781254, 4.84723158480412154608599408122, 5.24795006489362574006876923437, 6.60956909567927065497434824284, 7.24787274107521824847853311187, 7.979301539160255844409173997921, 9.042786150763052742751229253250, 9.857923215557343774523713570080
