L(s) = 1 | − 1.16·2-s + 2.40·3-s − 0.654·4-s + 5-s − 2.79·6-s + 1.39·7-s + 3.07·8-s + 2.78·9-s − 1.16·10-s − 3.58·11-s − 1.57·12-s + 5.88·13-s − 1.62·14-s + 2.40·15-s − 2.26·16-s − 3.58·17-s − 3.23·18-s + 6.25·19-s − 0.654·20-s + 3.36·21-s + 4.15·22-s − 3.36·23-s + 7.40·24-s + 25-s − 6.82·26-s − 0.517·27-s − 0.915·28-s + ⋯ |
L(s) = 1 | − 0.820·2-s + 1.38·3-s − 0.327·4-s + 0.447·5-s − 1.13·6-s + 0.528·7-s + 1.08·8-s + 0.928·9-s − 0.366·10-s − 1.08·11-s − 0.454·12-s + 1.63·13-s − 0.433·14-s + 0.621·15-s − 0.565·16-s − 0.870·17-s − 0.761·18-s + 1.43·19-s − 0.146·20-s + 0.734·21-s + 0.886·22-s − 0.701·23-s + 1.51·24-s + 0.200·25-s − 1.33·26-s − 0.0996·27-s − 0.172·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.814770022\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.814770022\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 + 1.16T + 2T^{2} \) |
| 3 | \( 1 - 2.40T + 3T^{2} \) |
| 7 | \( 1 - 1.39T + 7T^{2} \) |
| 11 | \( 1 + 3.58T + 11T^{2} \) |
| 13 | \( 1 - 5.88T + 13T^{2} \) |
| 17 | \( 1 + 3.58T + 17T^{2} \) |
| 19 | \( 1 - 6.25T + 19T^{2} \) |
| 23 | \( 1 + 3.36T + 23T^{2} \) |
| 29 | \( 1 - 6.90T + 29T^{2} \) |
| 31 | \( 1 - 8.83T + 31T^{2} \) |
| 37 | \( 1 + 1.87T + 37T^{2} \) |
| 41 | \( 1 + 3.58T + 41T^{2} \) |
| 43 | \( 1 + 8.14T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 - 6.43T + 53T^{2} \) |
| 59 | \( 1 + 6.67T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 3.73T + 67T^{2} \) |
| 71 | \( 1 + 9.13T + 71T^{2} \) |
| 73 | \( 1 - 3.23T + 73T^{2} \) |
| 79 | \( 1 - 3.99T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 8.15T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−9.593454523271409563053043896203, −8.720876428624188794415376033893, −8.305428573472159477711965294468, −7.84276878564019687850045723402, −6.71295081651890106000047038583, −5.42979308345700398461977450086, −4.45160785159099768723942398884, −3.37278169460781670227381017647, −2.30362190497589655542601524908, −1.18171845084036312215195853144,
1.18171845084036312215195853144, 2.30362190497589655542601524908, 3.37278169460781670227381017647, 4.45160785159099768723942398884, 5.42979308345700398461977450086, 6.71295081651890106000047038583, 7.84276878564019687850045723402, 8.305428573472159477711965294468, 8.720876428624188794415376033893, 9.593454523271409563053043896203