| L(s) = 1 | − 2-s + 4-s − 1.69·5-s − 0.423·7-s − 8-s + 1.69·10-s − 5.18·11-s + 1.72·13-s + 0.423·14-s + 16-s − 1.41·17-s − 4.68·19-s − 1.69·20-s + 5.18·22-s − 6.84·23-s − 2.11·25-s − 1.72·26-s − 0.423·28-s − 8.67·29-s + 0.736·31-s − 32-s + 1.41·34-s + 0.719·35-s − 11.0·37-s + 4.68·38-s + 1.69·40-s − 4.95·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.759·5-s − 0.160·7-s − 0.353·8-s + 0.537·10-s − 1.56·11-s + 0.478·13-s + 0.113·14-s + 0.250·16-s − 0.343·17-s − 1.07·19-s − 0.379·20-s + 1.10·22-s − 1.42·23-s − 0.422·25-s − 0.338·26-s − 0.0800·28-s − 1.61·29-s + 0.132·31-s − 0.176·32-s + 0.242·34-s + 0.121·35-s − 1.82·37-s + 0.760·38-s + 0.268·40-s − 0.773·41-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
| \(L(1)\) |
\(\approx\) |
\(0.2234747732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2234747732\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.69T + 5T^{2} \) |
| 7 | \( 1 + 0.423T + 7T^{2} \) |
| 11 | \( 1 + 5.18T + 11T^{2} \) |
| 13 | \( 1 - 1.72T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 + 4.68T + 19T^{2} \) |
| 23 | \( 1 + 6.84T + 23T^{2} \) |
| 29 | \( 1 + 8.67T + 29T^{2} \) |
| 31 | \( 1 - 0.736T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 + 4.95T + 41T^{2} \) |
| 43 | \( 1 - 9.88T + 43T^{2} \) |
| 47 | \( 1 - 4.67T + 47T^{2} \) |
| 53 | \( 1 - 6.11T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 9.62T + 61T^{2} \) |
| 67 | \( 1 + 4.22T + 67T^{2} \) |
| 71 | \( 1 - 6.48T + 71T^{2} \) |
| 73 | \( 1 + 5.24T + 73T^{2} \) |
| 79 | \( 1 - 7.76T + 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 5.04T + 97T^{2} \) |
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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
−7.83221859089575495214211143575, −7.44044920576899531436313171525, −6.55698882608410583871821369074, −5.83461597016730597324764131528, −5.14823367644581567903815156631, −4.08409649111857013611450727601, −3.55861871213784891795385431387, −2.46306578020368080171685204323, −1.82722834373533058116722416038, −0.24423216496667796374922099052,
0.24423216496667796374922099052, 1.82722834373533058116722416038, 2.46306578020368080171685204323, 3.55861871213784891795385431387, 4.08409649111857013611450727601, 5.14823367644581567903815156631, 5.83461597016730597324764131528, 6.55698882608410583871821369074, 7.44044920576899531436313171525, 7.83221859089575495214211143575
