L(s) = 1 | − 2-s − 1.61·3-s + 4-s + 1.61·6-s + 0.618·7-s − 8-s − 0.381·9-s − 2.85·11-s − 1.61·12-s + 7.09·13-s − 0.618·14-s + 16-s − 6.09·17-s + 0.381·18-s + 1.85·19-s − 1.00·21-s + 2.85·22-s − 23-s + 1.61·24-s − 7.09·26-s + 5.47·27-s + 0.618·28-s − 9.23·29-s + 9.09·31-s − 32-s + 4.61·33-s + 6.09·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.934·3-s + 0.5·4-s + 0.660·6-s + 0.233·7-s − 0.353·8-s − 0.127·9-s − 0.860·11-s − 0.467·12-s + 1.96·13-s − 0.165·14-s + 0.250·16-s − 1.47·17-s + 0.0900·18-s + 0.425·19-s − 0.218·21-s + 0.608·22-s − 0.208·23-s + 0.330·24-s − 1.39·26-s + 1.05·27-s + 0.116·28-s − 1.71·29-s + 1.63·31-s − 0.176·32-s + 0.803·33-s + 1.04·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + 1.61T + 3T^{2} \) |
| 7 | \( 1 - 0.618T + 7T^{2} \) |
| 11 | \( 1 + 2.85T + 11T^{2} \) |
| 13 | \( 1 - 7.09T + 13T^{2} \) |
| 17 | \( 1 + 6.09T + 17T^{2} \) |
| 19 | \( 1 - 1.85T + 19T^{2} \) |
| 29 | \( 1 + 9.23T + 29T^{2} \) |
| 31 | \( 1 - 9.09T + 31T^{2} \) |
| 37 | \( 1 + 6.47T + 37T^{2} \) |
| 41 | \( 1 - 3.32T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 3.70T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 - 1.70T + 59T^{2} \) |
| 61 | \( 1 + 9.32T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 4.09T + 71T^{2} \) |
| 73 | \( 1 + 3.23T + 73T^{2} \) |
| 79 | \( 1 - 1.52T + 79T^{2} \) |
| 83 | \( 1 - 6.94T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−9.295733110368483409577233262984, −8.582959192436207916173518428837, −7.901987380887807389085272989017, −6.77459977918644625875056233663, −6.07816567610664796363613436873, −5.37357833558689789638200397169, −4.19813928089961116308847248245, −2.87744013085417543521607280372, −1.46896043750359807264804449626, 0,
1.46896043750359807264804449626, 2.87744013085417543521607280372, 4.19813928089961116308847248245, 5.37357833558689789638200397169, 6.07816567610664796363613436873, 6.77459977918644625875056233663, 7.901987380887807389085272989017, 8.582959192436207916173518428837, 9.295733110368483409577233262984