| L(s) = 1 | + 0.924·2-s − 1.14·4-s + 5-s − 4.64·7-s − 2.90·8-s + 0.924·10-s − 4.29·14-s − 0.398·16-s − 6.37·17-s − 4.64·19-s − 1.14·20-s + 6.74·23-s + 25-s + 5.31·28-s − 3.46·29-s − 2.20·31-s + 5.44·32-s − 5.89·34-s − 4.64·35-s + 3.45·37-s − 4.29·38-s − 2.90·40-s − 5.81·41-s + 0.234·43-s + 6.23·46-s − 9.32·47-s + 14.5·49-s + ⋯ |
| L(s) = 1 | + 0.653·2-s − 0.572·4-s + 0.447·5-s − 1.75·7-s − 1.02·8-s + 0.292·10-s − 1.14·14-s − 0.0996·16-s − 1.54·17-s − 1.06·19-s − 0.256·20-s + 1.40·23-s + 0.200·25-s + 1.00·28-s − 0.643·29-s − 0.395·31-s + 0.962·32-s − 1.01·34-s − 0.784·35-s + 0.568·37-s − 0.695·38-s − 0.459·40-s − 0.908·41-s + 0.0357·43-s + 0.919·46-s − 1.36·47-s + 2.07·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.048263423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.048263423\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.924T + 2T^{2} \) |
| 7 | \( 1 + 4.64T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 6.37T + 17T^{2} \) |
| 19 | \( 1 + 4.64T + 19T^{2} \) |
| 23 | \( 1 - 6.74T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 2.20T + 31T^{2} \) |
| 37 | \( 1 - 3.45T + 37T^{2} \) |
| 41 | \( 1 + 5.81T + 41T^{2} \) |
| 43 | \( 1 - 0.234T + 43T^{2} \) |
| 47 | \( 1 + 9.32T + 47T^{2} \) |
| 53 | \( 1 - 3.54T + 53T^{2} \) |
| 59 | \( 1 - 14.5T + 59T^{2} \) |
| 61 | \( 1 - 1.96T + 61T^{2} \) |
| 67 | \( 1 + 1.45T + 67T^{2} \) |
| 71 | \( 1 + 2.58T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 - 3.58T + 79T^{2} \) |
| 83 | \( 1 + 7.16T + 83T^{2} \) |
| 89 | \( 1 + 8.98T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.502297132752761746327370082113, −7.03378790158809873230360161071, −6.64154237304694051745317563326, −6.02169886309471175245383813540, −5.26492585040811841408507566465, −4.45454545298235256182037830346, −3.71110077420174613090424793103, −3.01348719004933352911503119353, −2.18400224216568495254091073532, −0.46967146612338223587077934641,
0.46967146612338223587077934641, 2.18400224216568495254091073532, 3.01348719004933352911503119353, 3.71110077420174613090424793103, 4.45454545298235256182037830346, 5.26492585040811841408507566465, 6.02169886309471175245383813540, 6.64154237304694051745317563326, 7.03378790158809873230360161071, 8.502297132752761746327370082113
