| L(s) = 1 | − 0.716·2-s − 1.48·4-s − 1.54·5-s + 7-s + 2.49·8-s + 1.10·10-s + 0.301·13-s − 0.716·14-s + 1.18·16-s + 0.0579·17-s − 2.70·19-s + 2.29·20-s + 1.50·23-s − 2.61·25-s − 0.215·26-s − 1.48·28-s − 2.04·29-s + 4.94·31-s − 5.84·32-s − 0.0414·34-s − 1.54·35-s − 4.43·37-s + 1.93·38-s − 3.85·40-s + 2.69·41-s − 1.09·43-s − 1.07·46-s + ⋯ |
| L(s) = 1 | − 0.506·2-s − 0.743·4-s − 0.690·5-s + 0.377·7-s + 0.882·8-s + 0.349·10-s + 0.0835·13-s − 0.191·14-s + 0.296·16-s + 0.0140·17-s − 0.621·19-s + 0.513·20-s + 0.314·23-s − 0.523·25-s − 0.0422·26-s − 0.281·28-s − 0.378·29-s + 0.887·31-s − 1.03·32-s − 0.00711·34-s − 0.260·35-s − 0.728·37-s + 0.314·38-s − 0.609·40-s + 0.421·41-s − 0.167·43-s − 0.159·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.716T + 2T^{2} \) |
| 5 | \( 1 + 1.54T + 5T^{2} \) |
| 13 | \( 1 - 0.301T + 13T^{2} \) |
| 17 | \( 1 - 0.0579T + 17T^{2} \) |
| 19 | \( 1 + 2.70T + 19T^{2} \) |
| 23 | \( 1 - 1.50T + 23T^{2} \) |
| 29 | \( 1 + 2.04T + 29T^{2} \) |
| 31 | \( 1 - 4.94T + 31T^{2} \) |
| 37 | \( 1 + 4.43T + 37T^{2} \) |
| 41 | \( 1 - 2.69T + 41T^{2} \) |
| 43 | \( 1 + 1.09T + 43T^{2} \) |
| 47 | \( 1 - 7.68T + 47T^{2} \) |
| 53 | \( 1 - 7.18T + 53T^{2} \) |
| 59 | \( 1 - 0.0782T + 59T^{2} \) |
| 61 | \( 1 + 7.27T + 61T^{2} \) |
| 67 | \( 1 + 6.09T + 67T^{2} \) |
| 71 | \( 1 + 3.72T + 71T^{2} \) |
| 73 | \( 1 + 1.61T + 73T^{2} \) |
| 79 | \( 1 - 8.86T + 79T^{2} \) |
| 83 | \( 1 + 5.03T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 - 7.95T + 97T^{2} \) |
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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
−7.58874104855164357415280688201, −7.17583773085793603467482428995, −6.10961645948460157162000387604, −5.36735967878567197804115361543, −4.50871279120883716430308230213, −4.10739456519884007907891161832, −3.23709673988541168553427469671, −2.07484806875524805433500431572, −1.03604200360581724909929626856, 0,
1.03604200360581724909929626856, 2.07484806875524805433500431572, 3.23709673988541168553427469671, 4.10739456519884007907891161832, 4.50871279120883716430308230213, 5.36735967878567197804115361543, 6.10961645948460157162000387604, 7.17583773085793603467482428995, 7.58874104855164357415280688201
