| L(s) = 1 | − 2.40·2-s + 3.77·4-s + 3.30·7-s − 4.25·8-s + 1.99·11-s + 4.36·13-s − 7.94·14-s + 2.68·16-s + 5.44·17-s + 5.44·19-s − 4.78·22-s − 5.70·23-s − 10.4·26-s + 12.4·28-s + 8.98·29-s − 10.3·31-s + 2.06·32-s − 13.0·34-s + 37-s − 13.0·38-s − 3.00·41-s + 6.42·43-s + 7.51·44-s + 13.7·46-s + 5.15·47-s + 3.94·49-s + 16.4·52-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 1.88·4-s + 1.25·7-s − 1.50·8-s + 0.600·11-s + 1.21·13-s − 2.12·14-s + 0.670·16-s + 1.32·17-s + 1.24·19-s − 1.02·22-s − 1.18·23-s − 2.05·26-s + 2.35·28-s + 1.66·29-s − 1.86·31-s + 0.365·32-s − 2.24·34-s + 0.164·37-s − 2.12·38-s − 0.468·41-s + 0.980·43-s + 1.13·44-s + 2.02·46-s + 0.751·47-s + 0.563·49-s + 2.28·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.427342404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.427342404\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 + 2.40T + 2T^{2} \) |
| 7 | \( 1 - 3.30T + 7T^{2} \) |
| 11 | \( 1 - 1.99T + 11T^{2} \) |
| 13 | \( 1 - 4.36T + 13T^{2} \) |
| 17 | \( 1 - 5.44T + 17T^{2} \) |
| 19 | \( 1 - 5.44T + 19T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 - 8.98T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 41 | \( 1 + 3.00T + 41T^{2} \) |
| 43 | \( 1 - 6.42T + 43T^{2} \) |
| 47 | \( 1 - 5.15T + 47T^{2} \) |
| 53 | \( 1 - 9.95T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 4.46T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 0.905T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 4.55T + 83T^{2} \) |
| 89 | \( 1 - 0.207T + 89T^{2} \) |
| 97 | \( 1 - 4.54T + 97T^{2} \) |
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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
−7.966964696450066432180078684431, −7.48066207992693081295373335284, −6.70487899665146790304885965849, −5.88115029534407791619019261083, −5.24867132032071902532013912513, −4.13300593949893091992512530230, −3.34199138512767249045145119598, −2.18729920533042241532132883222, −1.36004306900996027443187643544, −0.900656146022360166750118730815,
0.900656146022360166750118730815, 1.36004306900996027443187643544, 2.18729920533042241532132883222, 3.34199138512767249045145119598, 4.13300593949893091992512530230, 5.24867132032071902532013912513, 5.88115029534407791619019261083, 6.70487899665146790304885965849, 7.48066207992693081295373335284, 7.966964696450066432180078684431
