| L(s) = 1 | + 2.32·3-s − 1.16·5-s + 2.40·9-s + 0.577·11-s − 2.18·13-s − 2.70·15-s − 6.65·17-s + 2.12·19-s + 8.32·23-s − 3.64·25-s − 1.37·27-s − 6.30·29-s + 1.38·31-s + 1.34·33-s + 2.86·37-s − 5.07·39-s + 41-s + 6.21·43-s − 2.80·45-s − 12.3·47-s − 15.4·51-s + 13.0·53-s − 0.672·55-s + 4.94·57-s + 3.27·59-s − 5.15·61-s + 2.54·65-s + ⋯ |
| L(s) = 1 | + 1.34·3-s − 0.520·5-s + 0.802·9-s + 0.174·11-s − 0.605·13-s − 0.699·15-s − 1.61·17-s + 0.488·19-s + 1.73·23-s − 0.728·25-s − 0.265·27-s − 1.16·29-s + 0.249·31-s + 0.233·33-s + 0.471·37-s − 0.812·39-s + 0.156·41-s + 0.947·43-s − 0.417·45-s − 1.80·47-s − 2.16·51-s + 1.79·53-s − 0.0907·55-s + 0.655·57-s + 0.426·59-s − 0.660·61-s + 0.315·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 - 2.32T + 3T^{2} \) |
| 5 | \( 1 + 1.16T + 5T^{2} \) |
| 11 | \( 1 - 0.577T + 11T^{2} \) |
| 13 | \( 1 + 2.18T + 13T^{2} \) |
| 17 | \( 1 + 6.65T + 17T^{2} \) |
| 19 | \( 1 - 2.12T + 19T^{2} \) |
| 23 | \( 1 - 8.32T + 23T^{2} \) |
| 29 | \( 1 + 6.30T + 29T^{2} \) |
| 31 | \( 1 - 1.38T + 31T^{2} \) |
| 37 | \( 1 - 2.86T + 37T^{2} \) |
| 43 | \( 1 - 6.21T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 - 3.27T + 59T^{2} \) |
| 61 | \( 1 + 5.15T + 61T^{2} \) |
| 67 | \( 1 + 5.35T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 + 8.30T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 6.56T + 89T^{2} \) |
| 97 | \( 1 - 2.09T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.33786388942525113239495202773, −7.27463255324622979922681024792, −6.22174684997794046153246862186, −5.26473507685037907030783448929, −4.40828415552190343989189401968, −3.87166564938981167402330378271, −2.97285277846121934176736287610, −2.46462551294407035629106272453, −1.48729833605377291927253909059, 0,
1.48729833605377291927253909059, 2.46462551294407035629106272453, 2.97285277846121934176736287610, 3.87166564938981167402330378271, 4.40828415552190343989189401968, 5.26473507685037907030783448929, 6.22174684997794046153246862186, 7.27463255324622979922681024792, 7.33786388942525113239495202773
