| L(s) = 1 | − 2.73·2-s − 3-s + 5.46·4-s + 5-s + 2.73·6-s − 9.46·8-s + 9-s − 2.73·10-s + 0.732·11-s − 5.46·12-s + 2.26·13-s − 15-s + 14.9·16-s + 3.26·17-s − 2.73·18-s + 4.46·19-s + 5.46·20-s − 2·22-s − 4.73·23-s + 9.46·24-s + 25-s − 6.19·26-s − 27-s − 4.19·29-s + 2.73·30-s − 0.464·31-s − 21.8·32-s + ⋯ |
| L(s) = 1 | − 1.93·2-s − 0.577·3-s + 2.73·4-s + 0.447·5-s + 1.11·6-s − 3.34·8-s + 0.333·9-s − 0.863·10-s + 0.220·11-s − 1.57·12-s + 0.629·13-s − 0.258·15-s + 3.73·16-s + 0.792·17-s − 0.643·18-s + 1.02·19-s + 1.22·20-s − 0.426·22-s − 0.986·23-s + 1.93·24-s + 0.200·25-s − 1.21·26-s − 0.192·27-s − 0.779·29-s + 0.498·30-s − 0.0833·31-s − 3.86·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5958002159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5958002159\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 11 | \( 1 - 0.732T + 11T^{2} \) |
| 13 | \( 1 - 2.26T + 13T^{2} \) |
| 17 | \( 1 - 3.26T + 17T^{2} \) |
| 19 | \( 1 - 4.46T + 19T^{2} \) |
| 23 | \( 1 + 4.73T + 23T^{2} \) |
| 29 | \( 1 + 4.19T + 29T^{2} \) |
| 31 | \( 1 + 0.464T + 31T^{2} \) |
| 37 | \( 1 + 3.19T + 37T^{2} \) |
| 41 | \( 1 + 0.732T + 41T^{2} \) |
| 43 | \( 1 - 3.19T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 + 0.196T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 - 6.19T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 7.39T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−10.22929369549912394541517448835, −9.569701230789221547532165236524, −8.837614841835761129453251935585, −7.86131857149675983830184212919, −7.16396310660547287368429365034, −6.19014696105195164971803590497, −5.51770006645377899765420329901, −3.49643118841849119941728596584, −2.02833646153387438431752175331, −0.907314652004903767442835188152,
0.907314652004903767442835188152, 2.02833646153387438431752175331, 3.49643118841849119941728596584, 5.51770006645377899765420329901, 6.19014696105195164971803590497, 7.16396310660547287368429365034, 7.86131857149675983830184212919, 8.837614841835761129453251935585, 9.569701230789221547532165236524, 10.22929369549912394541517448835
