| L(s) = 1 | + 1.34·2-s − 3-s − 0.196·4-s − 3.48·5-s − 1.34·6-s − 7-s − 2.94·8-s + 9-s − 4.68·10-s + 0.292·11-s + 0.196·12-s − 13-s − 1.34·14-s + 3.48·15-s − 3.56·16-s − 6.68·17-s + 1.34·18-s + 4.17·19-s + 0.685·20-s + 21-s + 0.393·22-s − 0.510·23-s + 2.94·24-s + 7.17·25-s − 1.34·26-s − 27-s + 0.196·28-s + ⋯ |
| L(s) = 1 | + 0.949·2-s − 0.577·3-s − 0.0982·4-s − 1.56·5-s − 0.548·6-s − 0.377·7-s − 1.04·8-s + 0.333·9-s − 1.48·10-s + 0.0882·11-s + 0.0567·12-s − 0.277·13-s − 0.358·14-s + 0.900·15-s − 0.892·16-s − 1.62·17-s + 0.316·18-s + 0.957·19-s + 0.153·20-s + 0.218·21-s + 0.0838·22-s − 0.106·23-s + 0.602·24-s + 1.43·25-s − 0.263·26-s − 0.192·27-s + 0.0371·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 1.34T + 2T^{2} \) |
| 5 | \( 1 + 3.48T + 5T^{2} \) |
| 11 | \( 1 - 0.292T + 11T^{2} \) |
| 17 | \( 1 + 6.68T + 17T^{2} \) |
| 19 | \( 1 - 4.17T + 19T^{2} \) |
| 23 | \( 1 + 0.510T + 23T^{2} \) |
| 29 | \( 1 - 6.17T + 29T^{2} \) |
| 31 | \( 1 + 5.78T + 31T^{2} \) |
| 37 | \( 1 + 0.978T + 37T^{2} \) |
| 41 | \( 1 - 0.685T + 41T^{2} \) |
| 43 | \( 1 - 1.19T + 43T^{2} \) |
| 47 | \( 1 + 7.19T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 + 8.97T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 + 7.27T + 71T^{2} \) |
| 73 | \( 1 - 6.76T + 73T^{2} \) |
| 79 | \( 1 + 2.80T + 79T^{2} \) |
| 83 | \( 1 - 6.17T + 83T^{2} \) |
| 89 | \( 1 + 9.88T + 89T^{2} \) |
| 97 | \( 1 + 2.61T + 97T^{2} \) |
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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
−11.66306847587123006321321965495, −10.94997439974829730867424835092, −9.511506926394470013522263309264, −8.472100238035409279247066236254, −7.26950254812755252054339121657, −6.31248303577370789161756290083, −4.95317750032561997593405383043, −4.22352194995039972389463480385, −3.17105547427276418647900894383, 0,
3.17105547427276418647900894383, 4.22352194995039972389463480385, 4.95317750032561997593405383043, 6.31248303577370789161756290083, 7.26950254812755252054339121657, 8.472100238035409279247066236254, 9.511506926394470013522263309264, 10.94997439974829730867424835092, 11.66306847587123006321321965495
