| L(s) = 1 | + 2.43·2-s + 3.90·4-s + 4.19·5-s − 3.26·7-s + 4.64·8-s + 10.1·10-s − 4.13·11-s + 5.15·13-s − 7.93·14-s + 3.46·16-s + 1.13·17-s + 0.476·19-s + 16.3·20-s − 10.0·22-s − 5.02·23-s + 12.5·25-s + 12.5·26-s − 12.7·28-s − 1.92·29-s + 5.49·31-s − 0.864·32-s + 2.75·34-s − 13.6·35-s − 10.1·37-s + 1.15·38-s + 19.4·40-s − 8.42·41-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 1.95·4-s + 1.87·5-s − 1.23·7-s + 1.64·8-s + 3.22·10-s − 1.24·11-s + 1.43·13-s − 2.12·14-s + 0.865·16-s + 0.274·17-s + 0.109·19-s + 3.66·20-s − 2.14·22-s − 1.04·23-s + 2.51·25-s + 2.45·26-s − 2.41·28-s − 0.357·29-s + 0.986·31-s − 0.152·32-s + 0.472·34-s − 2.31·35-s − 1.67·37-s + 0.188·38-s + 3.07·40-s − 1.31·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.905141333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.905141333\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 2.43T + 2T^{2} \) |
| 5 | \( 1 - 4.19T + 5T^{2} \) |
| 7 | \( 1 + 3.26T + 7T^{2} \) |
| 11 | \( 1 + 4.13T + 11T^{2} \) |
| 13 | \( 1 - 5.15T + 13T^{2} \) |
| 17 | \( 1 - 1.13T + 17T^{2} \) |
| 19 | \( 1 - 0.476T + 19T^{2} \) |
| 23 | \( 1 + 5.02T + 23T^{2} \) |
| 29 | \( 1 + 1.92T + 29T^{2} \) |
| 31 | \( 1 - 5.49T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 + 8.42T + 41T^{2} \) |
| 43 | \( 1 + 1.98T + 43T^{2} \) |
| 47 | \( 1 - 6.08T + 47T^{2} \) |
| 53 | \( 1 + 6.66T + 53T^{2} \) |
| 59 | \( 1 + 7.68T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 + 2.56T + 67T^{2} \) |
| 71 | \( 1 - 4.59T + 71T^{2} \) |
| 73 | \( 1 - 9.91T + 73T^{2} \) |
| 79 | \( 1 + 1.59T + 79T^{2} \) |
| 83 | \( 1 - 3.51T + 83T^{2} \) |
| 89 | \( 1 + 4.69T + 89T^{2} \) |
| 97 | \( 1 + 1.60T + 97T^{2} \) |
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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
−10.27097532220429289060080468367, −9.524446074960316570851927522455, −8.388096519275811239108065313002, −6.86017446192027049170873718145, −6.26767641096973330687596236689, −5.69474853117932269856413983861, −5.05646506032431928474606723356, −3.62174130282218495985890047485, −2.85684205754364261123070468636, −1.86551792825869039496606930863,
1.86551792825869039496606930863, 2.85684205754364261123070468636, 3.62174130282218495985890047485, 5.05646506032431928474606723356, 5.69474853117932269856413983861, 6.26767641096973330687596236689, 6.86017446192027049170873718145, 8.388096519275811239108065313002, 9.524446074960316570851927522455, 10.27097532220429289060080468367
