| L(s) = 1 | + 0.111·2-s + 1.47·3-s − 1.98·4-s + 5-s + 0.164·6-s − 0.779·7-s − 0.445·8-s − 0.832·9-s + 0.111·10-s + 5.49·11-s − 2.92·12-s − 0.334·13-s − 0.0872·14-s + 1.47·15-s + 3.92·16-s − 17-s − 0.0930·18-s − 0.109·19-s − 1.98·20-s − 1.14·21-s + 0.614·22-s − 8.37·23-s − 0.656·24-s + 25-s − 0.0373·26-s − 5.64·27-s + 1.54·28-s + ⋯ |
| L(s) = 1 | + 0.0790·2-s + 0.850·3-s − 0.993·4-s + 0.447·5-s + 0.0672·6-s − 0.294·7-s − 0.157·8-s − 0.277·9-s + 0.0353·10-s + 1.65·11-s − 0.844·12-s − 0.0927·13-s − 0.0233·14-s + 0.380·15-s + 0.981·16-s − 0.242·17-s − 0.0219·18-s − 0.0250·19-s − 0.444·20-s − 0.250·21-s + 0.131·22-s − 1.74·23-s − 0.134·24-s + 0.200·25-s − 0.00733·26-s − 1.08·27-s + 0.292·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| good | 2 | \( 1 - 0.111T + 2T^{2} \) |
| 3 | \( 1 - 1.47T + 3T^{2} \) |
| 7 | \( 1 + 0.779T + 7T^{2} \) |
| 11 | \( 1 - 5.49T + 11T^{2} \) |
| 13 | \( 1 + 0.334T + 13T^{2} \) |
| 19 | \( 1 + 0.109T + 19T^{2} \) |
| 23 | \( 1 + 8.37T + 23T^{2} \) |
| 29 | \( 1 + 7.91T + 29T^{2} \) |
| 31 | \( 1 + 0.807T + 31T^{2} \) |
| 37 | \( 1 + 2.19T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 2.95T + 43T^{2} \) |
| 47 | \( 1 + 1.21T + 47T^{2} \) |
| 53 | \( 1 + 0.341T + 53T^{2} \) |
| 59 | \( 1 - 2.85T + 59T^{2} \) |
| 61 | \( 1 + 7.16T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 73 | \( 1 + 2.49T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 0.0189T + 83T^{2} \) |
| 89 | \( 1 + 6.80T + 89T^{2} \) |
| 97 | \( 1 - 4.13T + 97T^{2} \) |
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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
−7.938900993755169130293861108780, −7.08739597445277278521001598585, −6.08427321938641572897536891812, −5.76298763903667128515882736028, −4.63636432669209541614596269268, −3.84565984918293151171282386815, −3.49544954640723256099647693621, −2.33045784272361683919795512505, −1.45084328469086187322107902394, 0,
1.45084328469086187322107902394, 2.33045784272361683919795512505, 3.49544954640723256099647693621, 3.84565984918293151171282386815, 4.63636432669209541614596269268, 5.76298763903667128515882736028, 6.08427321938641572897536891812, 7.08739597445277278521001598585, 7.938900993755169130293861108780
