| L(s) = 1 | + 2.36·2-s + 3.57·4-s − 3.93·5-s + 3.72·8-s − 9.29·10-s − 11-s + 3.93·13-s + 1.63·16-s + 4.72·17-s − 4.78·19-s − 14.0·20-s − 2.36·22-s − 2.72·23-s + 10.5·25-s + 9.29·26-s − 7.93·29-s − 1.15·31-s − 3.57·32-s + 11.1·34-s − 5.50·37-s − 11.2·38-s − 14.6·40-s + 0.430·41-s + 6.72·43-s − 3.57·44-s − 6.43·46-s − 8.78·47-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 1.78·4-s − 1.76·5-s + 1.31·8-s − 2.94·10-s − 0.301·11-s + 1.09·13-s + 0.409·16-s + 1.14·17-s − 1.09·19-s − 3.14·20-s − 0.503·22-s − 0.567·23-s + 2.10·25-s + 1.82·26-s − 1.47·29-s − 0.207·31-s − 0.632·32-s + 1.91·34-s − 0.905·37-s − 1.83·38-s − 2.31·40-s + 0.0671·41-s + 1.02·43-s − 0.539·44-s − 0.948·46-s − 1.28·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.36T + 2T^{2} \) |
| 5 | \( 1 + 3.93T + 5T^{2} \) |
| 13 | \( 1 - 3.93T + 13T^{2} \) |
| 17 | \( 1 - 4.72T + 17T^{2} \) |
| 19 | \( 1 + 4.78T + 19T^{2} \) |
| 23 | \( 1 + 2.72T + 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 + 1.15T + 31T^{2} \) |
| 37 | \( 1 + 5.50T + 37T^{2} \) |
| 41 | \( 1 - 0.430T + 41T^{2} \) |
| 43 | \( 1 - 6.72T + 43T^{2} \) |
| 47 | \( 1 + 8.78T + 47T^{2} \) |
| 53 | \( 1 - 3.15T + 53T^{2} \) |
| 59 | \( 1 + 15.0T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 3.21T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 + 5.44T + 79T^{2} \) |
| 83 | \( 1 + 2.84T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−7.66443206941579421518092226335, −7.17024162920674312725068779644, −6.20374071753601252736034225176, −5.66595855350543456443050258933, −4.70325819566239254849441120204, −4.11077705057673687655577701124, −3.54893493682141001499897370970, −3.00787552768999804729073298237, −1.66244486548185166315553131493, 0,
1.66244486548185166315553131493, 3.00787552768999804729073298237, 3.54893493682141001499897370970, 4.11077705057673687655577701124, 4.70325819566239254849441120204, 5.66595855350543456443050258933, 6.20374071753601252736034225176, 7.17024162920674312725068779644, 7.66443206941579421518092226335
