| L(s) = 1 | + 0.626·2-s − 1.60·4-s + 5-s + 4.43·7-s − 2.26·8-s + 0.626·10-s + 6.05·11-s + 2.77·14-s + 1.79·16-s − 5.84·17-s − 3.60·19-s − 1.60·20-s + 3.79·22-s + 2.26·23-s + 25-s − 7.12·28-s + 8.08·29-s + 6.45·31-s + 5.64·32-s − 3.66·34-s + 4.43·35-s − 1.79·37-s − 2.26·38-s − 2.26·40-s − 7.99·41-s + 6.48·43-s − 9.73·44-s + ⋯ |
| L(s) = 1 | + 0.443·2-s − 0.803·4-s + 0.447·5-s + 1.67·7-s − 0.799·8-s + 0.198·10-s + 1.82·11-s + 0.742·14-s + 0.449·16-s − 1.41·17-s − 0.827·19-s − 0.359·20-s + 0.809·22-s + 0.471·23-s + 0.200·25-s − 1.34·28-s + 1.50·29-s + 1.15·31-s + 0.998·32-s − 0.627·34-s + 0.749·35-s − 0.295·37-s − 0.366·38-s − 0.357·40-s − 1.24·41-s + 0.989·43-s − 1.46·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.152803802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.152803802\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.626T + 2T^{2} \) |
| 7 | \( 1 - 4.43T + 7T^{2} \) |
| 11 | \( 1 - 6.05T + 11T^{2} \) |
| 17 | \( 1 + 5.84T + 17T^{2} \) |
| 19 | \( 1 + 3.60T + 19T^{2} \) |
| 23 | \( 1 - 2.26T + 23T^{2} \) |
| 29 | \( 1 - 8.08T + 29T^{2} \) |
| 31 | \( 1 - 6.45T + 31T^{2} \) |
| 37 | \( 1 + 1.79T + 37T^{2} \) |
| 41 | \( 1 + 7.99T + 41T^{2} \) |
| 43 | \( 1 - 6.48T + 43T^{2} \) |
| 47 | \( 1 + 3.22T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 3.12T + 59T^{2} \) |
| 61 | \( 1 + 2.45T + 61T^{2} \) |
| 67 | \( 1 + 4.33T + 67T^{2} \) |
| 71 | \( 1 - 8.49T + 71T^{2} \) |
| 73 | \( 1 + 0.819T + 73T^{2} \) |
| 79 | \( 1 + 8.42T + 79T^{2} \) |
| 83 | \( 1 - 2.35T + 83T^{2} \) |
| 89 | \( 1 - 0.773T + 89T^{2} \) |
| 97 | \( 1 + 6.32T + 97T^{2} \) |
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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
−8.193903769372939094820719826760, −6.92005704505512403651578182659, −6.50631695652683856894646518502, −5.67785174479368351293646092860, −4.78517428367387874495306871777, −4.46928638090898614776760493647, −3.88795853516402148307989725995, −2.66913865729436401756597192896, −1.71640980442197004778416454081, −0.907027760340747847996422980849,
0.907027760340747847996422980849, 1.71640980442197004778416454081, 2.66913865729436401756597192896, 3.88795853516402148307989725995, 4.46928638090898614776760493647, 4.78517428367387874495306871777, 5.67785174479368351293646092860, 6.50631695652683856894646518502, 6.92005704505512403651578182659, 8.193903769372939094820719826760
