| L(s) = 1 | + 2-s + 4-s + 0.236·5-s + 7-s + 8-s + 0.236·10-s − 3.61·13-s + 14-s + 16-s − 0.618·17-s − 2.85·19-s + 0.236·20-s − 1.76·23-s − 4.94·25-s − 3.61·26-s + 28-s + 2.61·29-s − 4.70·31-s + 32-s − 0.618·34-s + 0.236·35-s + 1.85·37-s − 2.85·38-s + 0.236·40-s − 1.52·41-s − 4.14·43-s − 1.76·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.105·5-s + 0.377·7-s + 0.353·8-s + 0.0746·10-s − 1.00·13-s + 0.267·14-s + 0.250·16-s − 0.149·17-s − 0.654·19-s + 0.0527·20-s − 0.367·23-s − 0.988·25-s − 0.709·26-s + 0.188·28-s + 0.486·29-s − 0.845·31-s + 0.176·32-s − 0.105·34-s + 0.0399·35-s + 0.304·37-s − 0.462·38-s + 0.0373·40-s − 0.238·41-s − 0.632·43-s − 0.260·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 0.236T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 13 | \( 1 + 3.61T + 13T^{2} \) |
| 17 | \( 1 + 0.618T + 17T^{2} \) |
| 19 | \( 1 + 2.85T + 19T^{2} \) |
| 23 | \( 1 + 1.76T + 23T^{2} \) |
| 29 | \( 1 - 2.61T + 29T^{2} \) |
| 31 | \( 1 + 4.70T + 31T^{2} \) |
| 37 | \( 1 - 1.85T + 37T^{2} \) |
| 41 | \( 1 + 1.52T + 41T^{2} \) |
| 43 | \( 1 + 4.14T + 43T^{2} \) |
| 47 | \( 1 + 9.47T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 - 0.381T + 59T^{2} \) |
| 61 | \( 1 - 5.38T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 + 3.38T + 71T^{2} \) |
| 73 | \( 1 + 5.76T + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 - 0.763T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−7.76808016699328941058027110535, −6.72276757656664532028525555472, −6.31999932677306848608118360814, −5.32457683767629341217935994188, −4.85704730725849195964673228743, −4.08556455516120474223252733105, −3.26650054219761167947514152948, −2.30604731243885872588614428817, −1.64081779073701188596514978142, 0,
1.64081779073701188596514978142, 2.30604731243885872588614428817, 3.26650054219761167947514152948, 4.08556455516120474223252733105, 4.85704730725849195964673228743, 5.32457683767629341217935994188, 6.31999932677306848608118360814, 6.72276757656664532028525555472, 7.76808016699328941058027110535
