| L(s) = 1 | − 1.79·2-s + 3-s + 1.20·4-s − 1.79·6-s − 7-s + 1.41·8-s + 9-s − 11-s + 1.20·12-s − 13-s + 1.79·14-s − 4.95·16-s − 7.58·17-s − 1.79·18-s − 6.58·19-s − 21-s + 1.79·22-s + 5.58·23-s + 1.41·24-s + 1.79·26-s + 27-s − 1.20·28-s − 8.16·29-s + 3.58·31-s + 6.04·32-s − 33-s + 13.5·34-s + ⋯ |
| L(s) = 1 | − 1.26·2-s + 0.577·3-s + 0.604·4-s − 0.731·6-s − 0.377·7-s + 0.501·8-s + 0.333·9-s − 0.301·11-s + 0.348·12-s − 0.277·13-s + 0.478·14-s − 1.23·16-s − 1.83·17-s − 0.422·18-s − 1.51·19-s − 0.218·21-s + 0.381·22-s + 1.16·23-s + 0.289·24-s + 0.351·26-s + 0.192·27-s − 0.228·28-s − 1.51·29-s + 0.643·31-s + 1.06·32-s − 0.174·33-s + 2.32·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6450201269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6450201269\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 1.79T + 2T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 7.58T + 17T^{2} \) |
| 19 | \( 1 + 6.58T + 19T^{2} \) |
| 23 | \( 1 - 5.58T + 23T^{2} \) |
| 29 | \( 1 + 8.16T + 29T^{2} \) |
| 31 | \( 1 - 3.58T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 + 1.58T + 43T^{2} \) |
| 47 | \( 1 + 1.41T + 47T^{2} \) |
| 53 | \( 1 - 9.58T + 53T^{2} \) |
| 59 | \( 1 - 4.58T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 8.58T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 - 7.16T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 - 9.16T + 89T^{2} \) |
| 97 | \( 1 - 2.41T + 97T^{2} \) |
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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
−8.474488482482199247999781927299, −7.52451721303055805792350413210, −6.89888355717479577768646063098, −6.42760518470388660025804293608, −5.11266616425640761138173681536, −4.42901734059988168403680239493, −3.57215432253729357472842397096, −2.36800610064009426129857228553, −1.92154049140804054072943268368, −0.48105070852092504498161870187,
0.48105070852092504498161870187, 1.92154049140804054072943268368, 2.36800610064009426129857228553, 3.57215432253729357472842397096, 4.42901734059988168403680239493, 5.11266616425640761138173681536, 6.42760518470388660025804293608, 6.89888355717479577768646063098, 7.52451721303055805792350413210, 8.474488482482199247999781927299
