| L(s) = 1 | − 0.142·2-s − 1.97·4-s − 5-s − 1.55·7-s + 0.566·8-s + 0.142·10-s − 1.06·11-s − 6.73·13-s + 0.221·14-s + 3.87·16-s − 4.71·17-s + 3.03·19-s + 1.97·20-s + 0.151·22-s − 5.80·23-s + 25-s + 0.958·26-s + 3.08·28-s − 4.16·29-s − 8.70·31-s − 1.68·32-s + 0.670·34-s + 1.55·35-s + 6.57·37-s − 0.432·38-s − 0.566·40-s − 7.77·41-s + ⋯ |
| L(s) = 1 | − 0.100·2-s − 0.989·4-s − 0.447·5-s − 0.588·7-s + 0.200·8-s + 0.0449·10-s − 0.320·11-s − 1.86·13-s + 0.0591·14-s + 0.969·16-s − 1.14·17-s + 0.696·19-s + 0.442·20-s + 0.0322·22-s − 1.21·23-s + 0.200·25-s + 0.188·26-s + 0.582·28-s − 0.772·29-s − 1.56·31-s − 0.297·32-s + 0.114·34-s + 0.263·35-s + 1.08·37-s − 0.0700·38-s − 0.0895·40-s − 1.21·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2397815753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2397815753\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 107 | \( 1 + T \) |
| good | 2 | \( 1 + 0.142T + 2T^{2} \) |
| 7 | \( 1 + 1.55T + 7T^{2} \) |
| 11 | \( 1 + 1.06T + 11T^{2} \) |
| 13 | \( 1 + 6.73T + 13T^{2} \) |
| 17 | \( 1 + 4.71T + 17T^{2} \) |
| 19 | \( 1 - 3.03T + 19T^{2} \) |
| 23 | \( 1 + 5.80T + 23T^{2} \) |
| 29 | \( 1 + 4.16T + 29T^{2} \) |
| 31 | \( 1 + 8.70T + 31T^{2} \) |
| 37 | \( 1 - 6.57T + 37T^{2} \) |
| 41 | \( 1 + 7.77T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 1.17T + 53T^{2} \) |
| 59 | \( 1 - 3.58T + 59T^{2} \) |
| 61 | \( 1 - 0.168T + 61T^{2} \) |
| 67 | \( 1 - 0.691T + 67T^{2} \) |
| 71 | \( 1 + 6.38T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 + 0.977T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 - 19.4T + 97T^{2} \) |
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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
−8.240191007023400819172241162005, −7.56608096378751880888527293544, −7.08210328343596663492357780741, −5.99746471879040967071849798756, −5.22087328388911180434723543463, −4.56168168552649355680025996836, −3.86277873535409271151560793908, −2.96247704158286327951213002037, −1.94582216679540801204924583502, −0.26078318642860361067666544597,
0.26078318642860361067666544597, 1.94582216679540801204924583502, 2.96247704158286327951213002037, 3.86277873535409271151560793908, 4.56168168552649355680025996836, 5.22087328388911180434723543463, 5.99746471879040967071849798756, 7.08210328343596663492357780741, 7.56608096378751880888527293544, 8.240191007023400819172241162005
