| L(s) = 1 | − 0.618·2-s + 3-s − 1.61·4-s − 0.618·6-s + 0.236·7-s + 2.23·8-s + 9-s − 0.763·11-s − 1.61·12-s − 3.23·13-s − 0.145·14-s + 1.85·16-s − 6.47·17-s − 0.618·18-s + 19-s + 0.236·21-s + 0.472·22-s + 8.47·23-s + 2.23·24-s + 2.00·26-s + 27-s − 0.381·28-s − 9.47·29-s − 8·31-s − 5.61·32-s − 0.763·33-s + 4.00·34-s + ⋯ |
| L(s) = 1 | − 0.437·2-s + 0.577·3-s − 0.809·4-s − 0.252·6-s + 0.0892·7-s + 0.790·8-s + 0.333·9-s − 0.230·11-s − 0.467·12-s − 0.897·13-s − 0.0389·14-s + 0.463·16-s − 1.56·17-s − 0.145·18-s + 0.229·19-s + 0.0515·21-s + 0.100·22-s + 1.76·23-s + 0.456·24-s + 0.392·26-s + 0.192·27-s − 0.0721·28-s − 1.75·29-s − 1.43·31-s − 0.993·32-s − 0.132·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 7 | \( 1 - 0.236T + 7T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 23 | \( 1 - 8.47T + 23T^{2} \) |
| 29 | \( 1 + 9.47T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 4.76T + 37T^{2} \) |
| 41 | \( 1 - 1.47T + 41T^{2} \) |
| 43 | \( 1 - 12.9T + 43T^{2} \) |
| 47 | \( 1 - 5.23T + 47T^{2} \) |
| 53 | \( 1 + T + 53T^{2} \) |
| 59 | \( 1 + 6.70T + 59T^{2} \) |
| 61 | \( 1 + 7.47T + 61T^{2} \) |
| 67 | \( 1 + 3.70T + 67T^{2} \) |
| 71 | \( 1 + 1.29T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 + 7.70T + 83T^{2} \) |
| 89 | \( 1 + 5T + 89T^{2} \) |
| 97 | \( 1 + 3.70T + 97T^{2} \) |
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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
−9.181261381722060537884896846990, −8.601167175697993215128839140713, −7.41646744640811361960018894209, −7.23174901412969987237917317217, −5.67736581263467345663484127626, −4.80169256947718492348971663024, −4.05826716928951410698780780993, −2.87898590573163790275443095093, −1.68439519556190185103794196011, 0,
1.68439519556190185103794196011, 2.87898590573163790275443095093, 4.05826716928951410698780780993, 4.80169256947718492348971663024, 5.67736581263467345663484127626, 7.23174901412969987237917317217, 7.41646744640811361960018894209, 8.601167175697993215128839140713, 9.181261381722060537884896846990
