| L(s) = 1 | − 2-s − 3-s + 4-s + 2.06·5-s + 6-s + 1.71·7-s − 8-s + 9-s − 2.06·10-s + 2.42·11-s − 12-s − 1.83·13-s − 1.71·14-s − 2.06·15-s + 16-s + 17-s − 18-s + 6.51·19-s + 2.06·20-s − 1.71·21-s − 2.42·22-s − 8.25·23-s + 24-s − 0.746·25-s + 1.83·26-s − 27-s + 1.71·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.922·5-s + 0.408·6-s + 0.647·7-s − 0.353·8-s + 0.333·9-s − 0.652·10-s + 0.730·11-s − 0.288·12-s − 0.507·13-s − 0.457·14-s − 0.532·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 1.49·19-s + 0.461·20-s − 0.373·21-s − 0.516·22-s − 1.72·23-s + 0.204·24-s − 0.149·25-s + 0.358·26-s − 0.192·27-s + 0.323·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 - 2.06T + 5T^{2} \) |
| 7 | \( 1 - 1.71T + 7T^{2} \) |
| 11 | \( 1 - 2.42T + 11T^{2} \) |
| 13 | \( 1 + 1.83T + 13T^{2} \) |
| 19 | \( 1 - 6.51T + 19T^{2} \) |
| 23 | \( 1 + 8.25T + 23T^{2} \) |
| 29 | \( 1 + 3.10T + 29T^{2} \) |
| 31 | \( 1 + 8.19T + 31T^{2} \) |
| 37 | \( 1 + 2.81T + 37T^{2} \) |
| 41 | \( 1 - 1.97T + 41T^{2} \) |
| 43 | \( 1 + 7.41T + 43T^{2} \) |
| 47 | \( 1 - 1.00T + 47T^{2} \) |
| 53 | \( 1 + 3.55T + 53T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 16.1T + 67T^{2} \) |
| 71 | \( 1 + 1.89T + 71T^{2} \) |
| 73 | \( 1 - 0.948T + 73T^{2} \) |
| 79 | \( 1 + 7.92T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 + 8.67T + 89T^{2} \) |
| 97 | \( 1 + 4.88T + 97T^{2} \) |
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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
−7.57375124237520328600910353997, −7.22760252461007096788748598917, −6.17927939712901804891208789418, −5.74543929563725679535136561853, −5.07409959916282130260745804410, −4.08879551522319254213080968262, −3.08815077028944885375549270647, −1.78741377236415281815228907201, −1.53152370859834241047870482765, 0,
1.53152370859834241047870482765, 1.78741377236415281815228907201, 3.08815077028944885375549270647, 4.08879551522319254213080968262, 5.07409959916282130260745804410, 5.74543929563725679535136561853, 6.17927939712901804891208789418, 7.22760252461007096788748598917, 7.57375124237520328600910353997
