| L(s) = 1 | − 2·2-s + 2·4-s + 5-s + 3.60·7-s − 2·10-s − 3·11-s − 7.21·14-s − 4·16-s − 3.60·17-s − 7.21·19-s + 2·20-s + 6·22-s + 3.60·23-s + 25-s + 7.21·28-s − 7.21·29-s + 7.21·31-s + 8·32-s + 7.21·34-s + 3.60·35-s + 3.60·37-s + 14.4·38-s − 11·41-s + 4·43-s − 6·44-s − 7.21·46-s + 4·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s + 1.36·7-s − 0.632·10-s − 0.904·11-s − 1.92·14-s − 16-s − 0.874·17-s − 1.65·19-s + 0.447·20-s + 1.27·22-s + 0.751·23-s + 0.200·25-s + 1.36·28-s − 1.33·29-s + 1.29·31-s + 1.41·32-s + 1.23·34-s + 0.609·35-s + 0.592·37-s + 2.33·38-s − 1.71·41-s + 0.609·43-s − 0.904·44-s − 1.06·46-s + 0.583·47-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2T + 2T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 17 | \( 1 + 3.60T + 17T^{2} \) |
| 19 | \( 1 + 7.21T + 19T^{2} \) |
| 23 | \( 1 - 3.60T + 23T^{2} \) |
| 29 | \( 1 + 7.21T + 29T^{2} \) |
| 31 | \( 1 - 7.21T + 31T^{2} \) |
| 37 | \( 1 - 3.60T + 37T^{2} \) |
| 41 | \( 1 + 11T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 13T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 5T + 71T^{2} \) |
| 73 | \( 1 + 7.21T + 73T^{2} \) |
| 79 | \( 1 - 13T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + 3.60T + 97T^{2} \) |
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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
−7.81124073047152845656480205099, −7.05569899572073300451950979648, −6.42000379442205464605399726871, −5.40136591153822374794241899561, −4.76204396221889242899003029145, −4.08821157415317262881803808505, −2.51101442625650056711311197751, −2.08902286492655227701855372944, −1.19375967252927720352670646678, 0,
1.19375967252927720352670646678, 2.08902286492655227701855372944, 2.51101442625650056711311197751, 4.08821157415317262881803808505, 4.76204396221889242899003029145, 5.40136591153822374794241899561, 6.42000379442205464605399726871, 7.05569899572073300451950979648, 7.81124073047152845656480205099
