| L(s) = 1 | − 2·2-s − 3-s + 4·4-s − 17·5-s + 2·6-s + 35·7-s − 8·8-s − 26·9-s + 34·10-s − 2·11-s − 4·12-s − 70·14-s + 17·15-s + 16·16-s − 19·17-s + 52·18-s − 94·19-s − 68·20-s − 35·21-s + 4·22-s − 72·23-s + 8·24-s + 164·25-s + 53·27-s + 140·28-s + 246·29-s − 34·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.192·3-s + 1/2·4-s − 1.52·5-s + 0.136·6-s + 1.88·7-s − 0.353·8-s − 0.962·9-s + 1.07·10-s − 0.0548·11-s − 0.0962·12-s − 1.33·14-s + 0.292·15-s + 1/4·16-s − 0.271·17-s + 0.680·18-s − 1.13·19-s − 0.760·20-s − 0.363·21-s + 0.0387·22-s − 0.652·23-s + 0.0680·24-s + 1.31·25-s + 0.377·27-s + 0.944·28-s + 1.57·29-s − 0.206·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8929632291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8929632291\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + T + p^{3} T^{2} \) |
| 5 | \( 1 + 17 T + p^{3} T^{2} \) |
| 7 | \( 1 - 5 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 2 T + p^{3} T^{2} \) |
| 17 | \( 1 + 19 T + p^{3} T^{2} \) |
| 19 | \( 1 + 94 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 - 246 T + p^{3} T^{2} \) |
| 31 | \( 1 - 100 T + p^{3} T^{2} \) |
| 37 | \( 1 - 11 T + p^{3} T^{2} \) |
| 41 | \( 1 - 280 T + p^{3} T^{2} \) |
| 43 | \( 1 - 241 T + p^{3} T^{2} \) |
| 47 | \( 1 + 137 T + p^{3} T^{2} \) |
| 53 | \( 1 + 232 T + p^{3} T^{2} \) |
| 59 | \( 1 - 386 T + p^{3} T^{2} \) |
| 61 | \( 1 - 64 T + p^{3} T^{2} \) |
| 67 | \( 1 - 10 p T + p^{3} T^{2} \) |
| 71 | \( 1 + 55 T + p^{3} T^{2} \) |
| 73 | \( 1 - 838 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1016 T + p^{3} T^{2} \) |
| 83 | \( 1 + 420 T + p^{3} T^{2} \) |
| 89 | \( 1 - 934 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1154 T + p^{3} T^{2} \) |
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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
−11.16521591808865517324488136111, −10.55197208623142527031894422123, −8.843044632478939071120725902608, −8.140016339413773913030270770467, −7.83145317817579620664513948670, −6.46540155392417203572243620801, −5.01284097666047711110198894519, −4.07779472887992393518625022119, −2.39263887491999158052145943699, −0.71232094000727190740633156724,
0.71232094000727190740633156724, 2.39263887491999158052145943699, 4.07779472887992393518625022119, 5.01284097666047711110198894519, 6.46540155392417203572243620801, 7.83145317817579620664513948670, 8.140016339413773913030270770467, 8.843044632478939071120725902608, 10.55197208623142527031894422123, 11.16521591808865517324488136111
