| L(s) = 1 | − 2-s + 2.60·3-s − 4-s − 3.76·5-s − 2.60·6-s − 0.167·7-s + 3·8-s + 3.76·9-s + 3.76·10-s − 11-s − 2.60·12-s + 0.167·14-s − 9.80·15-s − 16-s + 4.37·17-s − 3.76·18-s + 3.37·19-s + 3.76·20-s − 0.434·21-s + 22-s − 5.20·23-s + 7.80·24-s + 9.20·25-s + 2.00·27-s + 0.167·28-s + 1.93·29-s + 9.80·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.50·3-s − 0.5·4-s − 1.68·5-s − 1.06·6-s − 0.0631·7-s + 1.06·8-s + 1.25·9-s + 1.19·10-s − 0.301·11-s − 0.751·12-s + 0.0446·14-s − 2.53·15-s − 0.250·16-s + 1.05·17-s − 0.888·18-s + 0.773·19-s + 0.842·20-s − 0.0948·21-s + 0.213·22-s − 1.08·23-s + 1.59·24-s + 1.84·25-s + 0.384·27-s + 0.0315·28-s + 0.359·29-s + 1.79·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + 2T^{2} \) |
| 3 | \( 1 - 2.60T + 3T^{2} \) |
| 5 | \( 1 + 3.76T + 5T^{2} \) |
| 7 | \( 1 + 0.167T + 7T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 - 3.37T + 19T^{2} \) |
| 23 | \( 1 + 5.20T + 23T^{2} \) |
| 29 | \( 1 - 1.93T + 29T^{2} \) |
| 31 | \( 1 + 6.86T + 31T^{2} \) |
| 37 | \( 1 + 8.63T + 37T^{2} \) |
| 41 | \( 1 - 1.26T + 41T^{2} \) |
| 43 | \( 1 + 7.20T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 - 1.33T + 53T^{2} \) |
| 59 | \( 1 + 2.13T + 59T^{2} \) |
| 61 | \( 1 + 4.80T + 61T^{2} \) |
| 67 | \( 1 - 4.13T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 + 3.56T + 97T^{2} \) |
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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
−8.538298757838176634606157430030, −8.300750352023258414043192083396, −7.48030421733173809924907587768, −7.27826251990656497401388546419, −5.43716698325643619821817027044, −4.34202510139997404983480576011, −3.67854344988793548825357380144, −3.04576354798957517936371072780, −1.52211600501641223530039251974, 0,
1.52211600501641223530039251974, 3.04576354798957517936371072780, 3.67854344988793548825357380144, 4.34202510139997404983480576011, 5.43716698325643619821817027044, 7.27826251990656497401388546419, 7.48030421733173809924907587768, 8.300750352023258414043192083396, 8.538298757838176634606157430030
