| L(s) = 1 | + 2.28·2-s + 1.22·3-s + 3.22·4-s − 2.50·5-s + 2.79·6-s − 4.50·7-s + 2.79·8-s − 1.50·9-s − 5.72·10-s − 11-s + 3.93·12-s − 10.2·14-s − 3.06·15-s − 0.0632·16-s − 3.22·17-s − 3.44·18-s + 2.34·19-s − 8.07·20-s − 5.50·21-s − 2.28·22-s + 5.79·23-s + 3.41·24-s + 1.28·25-s − 5.50·27-s − 14.5·28-s − 0.619·29-s − 7·30-s + ⋯ |
| L(s) = 1 | + 1.61·2-s + 0.705·3-s + 1.61·4-s − 1.12·5-s + 1.13·6-s − 1.70·7-s + 0.987·8-s − 0.502·9-s − 1.81·10-s − 0.301·11-s + 1.13·12-s − 2.75·14-s − 0.790·15-s − 0.0158·16-s − 0.781·17-s − 0.811·18-s + 0.538·19-s − 1.80·20-s − 1.20·21-s − 0.487·22-s + 1.20·23-s + 0.696·24-s + 0.257·25-s − 1.05·27-s − 2.74·28-s − 0.115·29-s − 1.27·30-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 | \( 1 - 2.28T + 2T^{2} \) |
| 3 | \( 1 - 1.22T + 3T^{2} \) |
| 5 | \( 1 + 2.50T + 5T^{2} \) |
| 7 | \( 1 + 4.50T + 7T^{2} \) |
| 17 | \( 1 + 3.22T + 17T^{2} \) |
| 19 | \( 1 - 2.34T + 19T^{2} \) |
| 23 | \( 1 - 5.79T + 23T^{2} \) |
| 29 | \( 1 + 0.619T + 29T^{2} \) |
| 31 | \( 1 + 6.01T + 31T^{2} \) |
| 37 | \( 1 + 5.06T + 37T^{2} \) |
| 41 | \( 1 - 8.36T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 6.74T + 47T^{2} \) |
| 53 | \( 1 + 4.06T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 0.144T + 61T^{2} \) |
| 67 | \( 1 + 8.72T + 67T^{2} \) |
| 71 | \( 1 - 1.71T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 1.87T + 79T^{2} \) |
| 83 | \( 1 - 5.38T + 83T^{2} \) |
| 89 | \( 1 - 8.50T + 89T^{2} \) |
| 97 | \( 1 + 0.112T + 97T^{2} \) |
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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
−9.029064852126175424701543541565, −7.75280016704423017239488884086, −7.13995672966184154119811162490, −6.31139480197183728210332762726, −5.55613804825973596348048282179, −4.50599636747869374283443191384, −3.61082324702614947203909139225, −3.21851029691190361073973325581, −2.47788355565979142086204206179, 0,
2.47788355565979142086204206179, 3.21851029691190361073973325581, 3.61082324702614947203909139225, 4.50599636747869374283443191384, 5.55613804825973596348048282179, 6.31139480197183728210332762726, 7.13995672966184154119811162490, 7.75280016704423017239488884086, 9.029064852126175424701543541565
