| L(s) = 1 | + 0.794·2-s − 1.36·4-s − 5-s + 3.59·7-s − 2.67·8-s − 0.794·10-s + 3.69·11-s + 2.85·14-s + 0.612·16-s + 7.56·17-s + 6.09·19-s + 1.36·20-s + 2.93·22-s + 4.84·23-s + 25-s − 4.92·28-s − 3.06·29-s + 5.25·31-s + 5.83·32-s + 6.00·34-s − 3.59·35-s − 4.14·37-s + 4.84·38-s + 2.67·40-s − 3.38·41-s − 2.62·43-s − 5.06·44-s + ⋯ |
| L(s) = 1 | + 0.561·2-s − 0.684·4-s − 0.447·5-s + 1.35·7-s − 0.946·8-s − 0.251·10-s + 1.11·11-s + 0.763·14-s + 0.153·16-s + 1.83·17-s + 1.39·19-s + 0.306·20-s + 0.626·22-s + 1.00·23-s + 0.200·25-s − 0.930·28-s − 0.569·29-s + 0.943·31-s + 1.03·32-s + 1.03·34-s − 0.608·35-s − 0.681·37-s + 0.785·38-s + 0.423·40-s − 0.529·41-s − 0.400·43-s − 0.763·44-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
| \(L(1)\) |
\(\approx\) |
\(2.994264893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.994264893\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.794T + 2T^{2} \) |
| 7 | \( 1 - 3.59T + 7T^{2} \) |
| 11 | \( 1 - 3.69T + 11T^{2} \) |
| 17 | \( 1 - 7.56T + 17T^{2} \) |
| 19 | \( 1 - 6.09T + 19T^{2} \) |
| 23 | \( 1 - 4.84T + 23T^{2} \) |
| 29 | \( 1 + 3.06T + 29T^{2} \) |
| 31 | \( 1 - 5.25T + 31T^{2} \) |
| 37 | \( 1 + 4.14T + 37T^{2} \) |
| 41 | \( 1 + 3.38T + 41T^{2} \) |
| 43 | \( 1 + 2.62T + 43T^{2} \) |
| 47 | \( 1 + 8.99T + 47T^{2} \) |
| 53 | \( 1 - 5.92T + 53T^{2} \) |
| 59 | \( 1 - 7.80T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 4.88T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 - 0.207T + 79T^{2} \) |
| 83 | \( 1 - 8.67T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−7.72395887307003304625354925076, −7.43584866653195654988007421455, −6.33531949198886131980520188953, −5.52113547446518191938326973856, −4.97723620270395459206604620507, −4.47383301050308755579945297707, −3.47638061677335771392905880606, −3.17495398248191436300854285500, −1.54687872431567189841284406898, −0.908118320950312397739518877311,
0.908118320950312397739518877311, 1.54687872431567189841284406898, 3.17495398248191436300854285500, 3.47638061677335771392905880606, 4.47383301050308755579945297707, 4.97723620270395459206604620507, 5.52113547446518191938326973856, 6.33531949198886131980520188953, 7.43584866653195654988007421455, 7.72395887307003304625354925076
