| L(s) = 1 | − 0.414·2-s − 1.82·4-s − 2·5-s + 1.58·8-s + 0.828·10-s − 11-s − 4.82·13-s + 3·16-s − 1.58·17-s + 1.24·19-s + 3.65·20-s + 0.414·22-s + 7·23-s − 25-s + 1.99·26-s + 5.24·29-s + 5.65·31-s − 4.41·32-s + 0.656·34-s + 7.48·37-s − 0.514·38-s − 3.17·40-s − 6.82·41-s − 11.2·43-s + 1.82·44-s − 2.89·46-s + 4.17·47-s + ⋯ |
| L(s) = 1 | − 0.292·2-s − 0.914·4-s − 0.894·5-s + 0.560·8-s + 0.261·10-s − 0.301·11-s − 1.33·13-s + 0.750·16-s − 0.384·17-s + 0.285·19-s + 0.817·20-s + 0.0883·22-s + 1.45·23-s − 0.200·25-s + 0.392·26-s + 0.973·29-s + 1.01·31-s − 0.780·32-s + 0.112·34-s + 1.23·37-s − 0.0834·38-s − 0.501·40-s − 1.06·41-s − 1.71·43-s + 0.275·44-s − 0.427·46-s + 0.608·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 + 1.58T + 17T^{2} \) |
| 19 | \( 1 - 1.24T + 19T^{2} \) |
| 23 | \( 1 - 7T + 23T^{2} \) |
| 29 | \( 1 - 5.24T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 - 7.48T + 37T^{2} \) |
| 41 | \( 1 + 6.82T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 4.17T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 - 2.65T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 8.82T + 67T^{2} \) |
| 71 | \( 1 - 9.82T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 9.31T + 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 5.48T + 97T^{2} \) |
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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
−8.013310597149278640594497958503, −7.32874370818733526107216500160, −6.72338240268617345048386194246, −5.48365316375383930385240237846, −4.80617143908473956648464794350, −4.34628880272666651491357226532, −3.35874488424920387228680921399, −2.51037583998353761264740630262, −1.01277526789946639591207650809, 0,
1.01277526789946639591207650809, 2.51037583998353761264740630262, 3.35874488424920387228680921399, 4.34628880272666651491357226532, 4.80617143908473956648464794350, 5.48365316375383930385240237846, 6.72338240268617345048386194246, 7.32874370818733526107216500160, 8.013310597149278640594497958503
