| L(s) = 1 | − 2-s + 4-s + 0.859·5-s − 4.94·7-s − 8-s − 0.859·10-s − 5.12·11-s − 5.12·13-s + 4.94·14-s + 16-s + 4.90·17-s − 3.91·19-s + 0.859·20-s + 5.12·22-s − 5.07·23-s − 4.26·25-s + 5.12·26-s − 4.94·28-s − 3.74·29-s − 6.15·31-s − 32-s − 4.90·34-s − 4.25·35-s + 0.392·37-s + 3.91·38-s − 0.859·40-s − 6.09·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.384·5-s − 1.87·7-s − 0.353·8-s − 0.271·10-s − 1.54·11-s − 1.42·13-s + 1.32·14-s + 0.250·16-s + 1.18·17-s − 0.898·19-s + 0.192·20-s + 1.09·22-s − 1.05·23-s − 0.852·25-s + 1.00·26-s − 0.935·28-s − 0.695·29-s − 1.10·31-s − 0.176·32-s − 0.841·34-s − 0.719·35-s + 0.0644·37-s + 0.635·38-s − 0.135·40-s − 0.951·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02467476839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02467476839\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 - 0.859T + 5T^{2} \) |
| 7 | \( 1 + 4.94T + 7T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 - 4.90T + 17T^{2} \) |
| 19 | \( 1 + 3.91T + 19T^{2} \) |
| 23 | \( 1 + 5.07T + 23T^{2} \) |
| 29 | \( 1 + 3.74T + 29T^{2} \) |
| 31 | \( 1 + 6.15T + 31T^{2} \) |
| 37 | \( 1 - 0.392T + 37T^{2} \) |
| 41 | \( 1 + 6.09T + 41T^{2} \) |
| 43 | \( 1 - 2.37T + 43T^{2} \) |
| 47 | \( 1 - 1.68T + 47T^{2} \) |
| 53 | \( 1 + 9.20T + 53T^{2} \) |
| 59 | \( 1 - 5.63T + 59T^{2} \) |
| 61 | \( 1 - 4.25T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 - 3.44T + 73T^{2} \) |
| 79 | \( 1 + 0.104T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 3.19T + 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−7.64681014060837062115108041551, −7.36525059791705862121130905856, −6.51352896849326865034620021734, −5.74467489440208267664556943255, −5.42155735615600332540232547953, −4.15175842834423592390969498399, −3.21598655132216850703178088311, −2.62800326302420674257688889144, −1.87862273346800775447338749578, −0.07878969983749651164462390856,
0.07878969983749651164462390856, 1.87862273346800775447338749578, 2.62800326302420674257688889144, 3.21598655132216850703178088311, 4.15175842834423592390969498399, 5.42155735615600332540232547953, 5.74467489440208267664556943255, 6.51352896849326865034620021734, 7.36525059791705862121130905856, 7.64681014060837062115108041551
