| L(s) = 1 | + 0.149·2-s + 2.76·3-s − 1.97·4-s − 4.13·5-s + 0.413·6-s − 7-s − 0.595·8-s + 4.61·9-s − 0.618·10-s + 2.55·11-s − 5.45·12-s − 0.149·14-s − 11.4·15-s + 3.86·16-s + 1.50·17-s + 0.691·18-s + 5.93·19-s + 8.17·20-s − 2.76·21-s + 0.382·22-s + 6.55·23-s − 1.64·24-s + 12.0·25-s + 4.46·27-s + 1.97·28-s + 0.283·29-s − 1.70·30-s + ⋯ |
| L(s) = 1 | + 0.105·2-s + 1.59·3-s − 0.988·4-s − 1.84·5-s + 0.168·6-s − 0.377·7-s − 0.210·8-s + 1.53·9-s − 0.195·10-s + 0.769·11-s − 1.57·12-s − 0.0399·14-s − 2.94·15-s + 0.966·16-s + 0.364·17-s + 0.162·18-s + 1.36·19-s + 1.82·20-s − 0.602·21-s + 0.0814·22-s + 1.36·23-s − 0.335·24-s + 2.41·25-s + 0.860·27-s + 0.373·28-s + 0.0527·29-s − 0.311·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.758427810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.758427810\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.149T + 2T^{2} \) |
| 3 | \( 1 - 2.76T + 3T^{2} \) |
| 5 | \( 1 + 4.13T + 5T^{2} \) |
| 11 | \( 1 - 2.55T + 11T^{2} \) |
| 17 | \( 1 - 1.50T + 17T^{2} \) |
| 19 | \( 1 - 5.93T + 19T^{2} \) |
| 23 | \( 1 - 6.55T + 23T^{2} \) |
| 29 | \( 1 - 0.283T + 29T^{2} \) |
| 31 | \( 1 - 1.95T + 31T^{2} \) |
| 37 | \( 1 + 5.66T + 37T^{2} \) |
| 41 | \( 1 - 6.70T + 41T^{2} \) |
| 43 | \( 1 + 8.14T + 43T^{2} \) |
| 47 | \( 1 - 3.94T + 47T^{2} \) |
| 53 | \( 1 + 1.08T + 53T^{2} \) |
| 59 | \( 1 - 3.71T + 59T^{2} \) |
| 61 | \( 1 - 1.93T + 61T^{2} \) |
| 67 | \( 1 - 3.38T + 67T^{2} \) |
| 71 | \( 1 + 5.36T + 71T^{2} \) |
| 73 | \( 1 + 2.62T + 73T^{2} \) |
| 79 | \( 1 - 7.89T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 6.64T + 89T^{2} \) |
| 97 | \( 1 + 0.504T + 97T^{2} \) |
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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
−9.317158971075023343011103339287, −8.956487614950046803556327869830, −8.176142288829424926457277148495, −7.60389045374620410994657412138, −6.86907709970245848869108671933, −5.13055424422571020159581583997, −4.19920239640920815122239542158, −3.49590640023676070300476344939, −3.07904917687681866656038205680, −0.956487977235816011426492607144,
0.956487977235816011426492607144, 3.07904917687681866656038205680, 3.49590640023676070300476344939, 4.19920239640920815122239542158, 5.13055424422571020159581583997, 6.86907709970245848869108671933, 7.60389045374620410994657412138, 8.176142288829424926457277148495, 8.956487614950046803556327869830, 9.317158971075023343011103339287
