L(s) = 1 | + 1.14·2-s − 0.361·3-s − 0.678·4-s − 0.415·6-s + 3.63·7-s − 3.07·8-s − 2.86·9-s − 3.31·11-s + 0.245·12-s − 3.42·13-s + 4.18·14-s − 2.18·16-s + 0.605·17-s − 3.29·18-s − 1.21·19-s − 1.31·21-s − 3.80·22-s − 6.22·23-s + 1.11·24-s − 3.93·26-s + 2.12·27-s − 2.46·28-s + 3.29·29-s + 0.187·31-s + 3.65·32-s + 1.19·33-s + 0.696·34-s + ⋯ |
L(s) = 1 | + 0.812·2-s − 0.208·3-s − 0.339·4-s − 0.169·6-s + 1.37·7-s − 1.08·8-s − 0.956·9-s − 0.998·11-s + 0.0708·12-s − 0.949·13-s + 1.11·14-s − 0.545·16-s + 0.146·17-s − 0.777·18-s − 0.279·19-s − 0.286·21-s − 0.811·22-s − 1.29·23-s + 0.227·24-s − 0.771·26-s + 0.408·27-s − 0.466·28-s + 0.611·29-s + 0.0335·31-s + 0.645·32-s + 0.208·33-s + 0.119·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.680637383\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.680637383\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 - 1.14T + 2T^{2} \) |
| 3 | \( 1 + 0.361T + 3T^{2} \) |
| 7 | \( 1 - 3.63T + 7T^{2} \) |
| 11 | \( 1 + 3.31T + 11T^{2} \) |
| 13 | \( 1 + 3.42T + 13T^{2} \) |
| 17 | \( 1 - 0.605T + 17T^{2} \) |
| 19 | \( 1 + 1.21T + 19T^{2} \) |
| 23 | \( 1 + 6.22T + 23T^{2} \) |
| 29 | \( 1 - 3.29T + 29T^{2} \) |
| 31 | \( 1 - 0.187T + 31T^{2} \) |
| 37 | \( 1 - 0.174T + 37T^{2} \) |
| 41 | \( 1 - 9.28T + 41T^{2} \) |
| 43 | \( 1 - 5.81T + 43T^{2} \) |
| 47 | \( 1 - 3.12T + 47T^{2} \) |
| 53 | \( 1 + 2.23T + 53T^{2} \) |
| 59 | \( 1 - 6.04T + 59T^{2} \) |
| 61 | \( 1 - 2.39T + 61T^{2} \) |
| 67 | \( 1 + 5.48T + 67T^{2} \) |
| 71 | \( 1 - 6.38T + 71T^{2} \) |
| 73 | \( 1 - 0.625T + 73T^{2} \) |
| 79 | \( 1 - 6.61T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 + 1.26T + 89T^{2} \) |
| 97 | \( 1 + 0.489T + 97T^{2} \) |
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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
−7.997207978613697702664077677312, −7.58212628018581154176703294700, −6.34387022441655903034036857780, −5.67835004149286792584298120083, −5.13258540177992853972807434766, −4.62794485468658224390508422325, −3.87170923069190620923008851173, −2.73301268440412714102416905923, −2.20455719588506214142461662420, −0.58347083159948251403390761609,
0.58347083159948251403390761609, 2.20455719588506214142461662420, 2.73301268440412714102416905923, 3.87170923069190620923008851173, 4.62794485468658224390508422325, 5.13258540177992853972807434766, 5.67835004149286792584298120083, 6.34387022441655903034036857780, 7.58212628018581154176703294700, 7.997207978613697702664077677312