L(s) = 1 | + 2-s − 3.08·3-s + 4-s − 5-s − 3.08·6-s − 2.62·7-s + 8-s + 6.50·9-s − 10-s + 0.190·11-s − 3.08·12-s − 13-s − 2.62·14-s + 3.08·15-s + 16-s − 6.93·17-s + 6.50·18-s − 0.237·19-s − 20-s + 8.09·21-s + 0.190·22-s + 4.00·23-s − 3.08·24-s + 25-s − 26-s − 10.7·27-s − 2.62·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.77·3-s + 0.5·4-s − 0.447·5-s − 1.25·6-s − 0.992·7-s + 0.353·8-s + 2.16·9-s − 0.316·10-s + 0.0573·11-s − 0.889·12-s − 0.277·13-s − 0.701·14-s + 0.795·15-s + 0.250·16-s − 1.68·17-s + 1.53·18-s − 0.0544·19-s − 0.223·20-s + 1.76·21-s + 0.0405·22-s + 0.835·23-s − 0.629·24-s + 0.200·25-s − 0.196·26-s − 2.07·27-s − 0.496·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7189092777\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7189092777\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 + 3.08T + 3T^{2} \) |
| 7 | \( 1 + 2.62T + 7T^{2} \) |
| 11 | \( 1 - 0.190T + 11T^{2} \) |
| 17 | \( 1 + 6.93T + 17T^{2} \) |
| 19 | \( 1 + 0.237T + 19T^{2} \) |
| 23 | \( 1 - 4.00T + 23T^{2} \) |
| 29 | \( 1 - 0.0461T + 29T^{2} \) |
| 37 | \( 1 + 9.02T + 37T^{2} \) |
| 41 | \( 1 + 2.50T + 41T^{2} \) |
| 43 | \( 1 + 2.72T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 1.99T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 0.753T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 6.14T + 71T^{2} \) |
| 73 | \( 1 + 5.91T + 73T^{2} \) |
| 79 | \( 1 - 0.817T + 79T^{2} \) |
| 83 | \( 1 - 0.435T + 83T^{2} \) |
| 89 | \( 1 + 1.09T + 89T^{2} \) |
| 97 | \( 1 - 8.17T + 97T^{2} \) |
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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
−8.357418008609312900637231586746, −7.10433163817356466791134675739, −6.76627781021298076786751457837, −6.32534813072699113273958828252, −5.33851195169697819947305796181, −4.86753671725306544566738345874, −4.07699460132144473485264224153, −3.22320382963540322605273198910, −1.90715147558623358983163568697, −0.46056767795535108364305203318,
0.46056767795535108364305203318, 1.90715147558623358983163568697, 3.22320382963540322605273198910, 4.07699460132144473485264224153, 4.86753671725306544566738345874, 5.33851195169697819947305796181, 6.32534813072699113273958828252, 6.76627781021298076786751457837, 7.10433163817356466791134675739, 8.357418008609312900637231586746