| L(s) = 1 | + 2.43·2-s + 1.52·3-s + 3.92·4-s + 3.49·5-s + 3.70·6-s − 4.87·7-s + 4.69·8-s − 0.678·9-s + 8.51·10-s − 1.63·11-s + 5.98·12-s − 5.07·13-s − 11.8·14-s + 5.32·15-s + 3.58·16-s + 7.41·17-s − 1.65·18-s + 0.289·19-s + 13.7·20-s − 7.42·21-s − 3.97·22-s − 23-s + 7.15·24-s + 7.21·25-s − 12.3·26-s − 5.60·27-s − 19.1·28-s + ⋯ |
| L(s) = 1 | + 1.72·2-s + 0.879·3-s + 1.96·4-s + 1.56·5-s + 1.51·6-s − 1.84·7-s + 1.66·8-s − 0.226·9-s + 2.69·10-s − 0.491·11-s + 1.72·12-s − 1.40·13-s − 3.17·14-s + 1.37·15-s + 0.895·16-s + 1.79·17-s − 0.389·18-s + 0.0665·19-s + 3.07·20-s − 1.61·21-s − 0.846·22-s − 0.208·23-s + 1.46·24-s + 1.44·25-s − 2.42·26-s − 1.07·27-s − 3.61·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.033285760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.033285760\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 2.43T + 2T^{2} \) |
| 3 | \( 1 - 1.52T + 3T^{2} \) |
| 5 | \( 1 - 3.49T + 5T^{2} \) |
| 7 | \( 1 + 4.87T + 7T^{2} \) |
| 11 | \( 1 + 1.63T + 11T^{2} \) |
| 13 | \( 1 + 5.07T + 13T^{2} \) |
| 17 | \( 1 - 7.41T + 17T^{2} \) |
| 19 | \( 1 - 0.289T + 19T^{2} \) |
| 31 | \( 1 + 1.20T + 31T^{2} \) |
| 37 | \( 1 - 2.03T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 5.66T + 43T^{2} \) |
| 47 | \( 1 - 5.12T + 47T^{2} \) |
| 53 | \( 1 - 7.16T + 53T^{2} \) |
| 59 | \( 1 + 0.758T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 3.60T + 67T^{2} \) |
| 71 | \( 1 - 8.89T + 71T^{2} \) |
| 73 | \( 1 + 2.62T + 73T^{2} \) |
| 79 | \( 1 + 3.77T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 4.54T + 89T^{2} \) |
| 97 | \( 1 + 2.62T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.12640293022437506810511157118, −9.972277304518401860281897175846, −9.027411875249325964181257127421, −7.50137109933417510664583948996, −6.62901940036648995157423664362, −5.71462616193219501430814597889, −5.31265508702563736662560306949, −3.67301161995692713364833682882, −2.85582297696186096288780121729, −2.34553721257791121360180092001,
2.34553721257791121360180092001, 2.85582297696186096288780121729, 3.67301161995692713364833682882, 5.31265508702563736662560306949, 5.71462616193219501430814597889, 6.62901940036648995157423664362, 7.50137109933417510664583948996, 9.027411875249325964181257127421, 9.972277304518401860281897175846, 10.12640293022437506810511157118
