| L(s) = 1 | + 2-s + 4-s − 5-s + 0.236·7-s + 8-s − 10-s − 4.61·13-s + 0.236·14-s + 16-s + 2.38·17-s + 1.38·19-s − 20-s − 3.47·23-s − 4·25-s − 4.61·26-s + 0.236·28-s − 4.09·29-s + 7.47·31-s + 32-s + 2.38·34-s − 0.236·35-s + 0.381·37-s + 1.38·38-s − 40-s + 6.47·41-s + 2.09·43-s − 3.47·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.0892·7-s + 0.353·8-s − 0.316·10-s − 1.28·13-s + 0.0630·14-s + 0.250·16-s + 0.577·17-s + 0.317·19-s − 0.223·20-s − 0.723·23-s − 0.800·25-s − 0.905·26-s + 0.0446·28-s − 0.759·29-s + 1.34·31-s + 0.176·32-s + 0.408·34-s − 0.0399·35-s + 0.0627·37-s + 0.224·38-s − 0.158·40-s + 1.01·41-s + 0.318·43-s − 0.511·46-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 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 - 0.236T + 7T^{2} \) |
| 13 | \( 1 + 4.61T + 13T^{2} \) |
| 17 | \( 1 - 2.38T + 17T^{2} \) |
| 19 | \( 1 - 1.38T + 19T^{2} \) |
| 23 | \( 1 + 3.47T + 23T^{2} \) |
| 29 | \( 1 + 4.09T + 29T^{2} \) |
| 31 | \( 1 - 7.47T + 31T^{2} \) |
| 37 | \( 1 - 0.381T + 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 - 2.09T + 43T^{2} \) |
| 47 | \( 1 + 1.76T + 47T^{2} \) |
| 53 | \( 1 - 3.85T + 53T^{2} \) |
| 59 | \( 1 + 8.14T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 + 1.29T + 67T^{2} \) |
| 71 | \( 1 + 7.61T + 71T^{2} \) |
| 73 | \( 1 + 0.236T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−7.73134143922915761250807284381, −6.98575017093639238781395506767, −6.08619960894202331388552236787, −5.52791860699639339986526201468, −4.63529673008436807368663548042, −4.17012632676567612767020148072, −3.20314303799194841537502848435, −2.51488653412919015829267201844, −1.47152172796995055819914139203, 0,
1.47152172796995055819914139203, 2.51488653412919015829267201844, 3.20314303799194841537502848435, 4.17012632676567612767020148072, 4.63529673008436807368663548042, 5.52791860699639339986526201468, 6.08619960894202331388552236787, 6.98575017093639238781395506767, 7.73134143922915761250807284381
