| L(s) = 1 | − 2.06·2-s + 1.49·3-s + 2.27·4-s − 5-s − 3.08·6-s − 2.20·7-s − 0.571·8-s − 0.775·9-s + 2.06·10-s + 0.732·11-s + 3.39·12-s + 0.652·13-s + 4.56·14-s − 1.49·15-s − 3.37·16-s − 0.343·17-s + 1.60·18-s + 4.07·19-s − 2.27·20-s − 3.28·21-s − 1.51·22-s + 7.46·23-s − 0.852·24-s + 25-s − 1.34·26-s − 5.63·27-s − 5.02·28-s + ⋯ |
| L(s) = 1 | − 1.46·2-s + 0.861·3-s + 1.13·4-s − 0.447·5-s − 1.25·6-s − 0.833·7-s − 0.202·8-s − 0.258·9-s + 0.653·10-s + 0.220·11-s + 0.980·12-s + 0.180·13-s + 1.21·14-s − 0.385·15-s − 0.842·16-s − 0.0832·17-s + 0.378·18-s + 0.935·19-s − 0.509·20-s − 0.717·21-s − 0.322·22-s + 1.55·23-s − 0.174·24-s + 0.200·25-s − 0.264·26-s − 1.08·27-s − 0.948·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 2.06T + 2T^{2} \) |
| 3 | \( 1 - 1.49T + 3T^{2} \) |
| 7 | \( 1 + 2.20T + 7T^{2} \) |
| 11 | \( 1 - 0.732T + 11T^{2} \) |
| 13 | \( 1 - 0.652T + 13T^{2} \) |
| 17 | \( 1 + 0.343T + 17T^{2} \) |
| 19 | \( 1 - 4.07T + 19T^{2} \) |
| 23 | \( 1 - 7.46T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 + 1.82T + 31T^{2} \) |
| 37 | \( 1 + 5.87T + 37T^{2} \) |
| 41 | \( 1 + 7.34T + 41T^{2} \) |
| 43 | \( 1 + 2.73T + 43T^{2} \) |
| 47 | \( 1 - 0.681T + 47T^{2} \) |
| 53 | \( 1 - 3.55T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 9.05T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 8.85T + 73T^{2} \) |
| 79 | \( 1 - 4.65T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 + 0.646T + 89T^{2} \) |
| 97 | \( 1 - 8.57T + 97T^{2} \) |
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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
−9.030365876710647626060806248497, −8.898333705428571025349882375972, −7.79723891993743581599888654263, −7.30962621779074178160089176142, −6.40688463257493842943004583784, −5.08811959970313950235932837657, −3.61737192781852640062012145191, −2.92333814065363420984688023551, −1.55673862614501803913458815138, 0,
1.55673862614501803913458815138, 2.92333814065363420984688023551, 3.61737192781852640062012145191, 5.08811959970313950235932837657, 6.40688463257493842943004583784, 7.30962621779074178160089176142, 7.79723891993743581599888654263, 8.898333705428571025349882375972, 9.030365876710647626060806248497
