L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 5.13·7-s + 8-s + 9-s + 10-s + 2.22·11-s + 12-s − 6.60·13-s + 5.13·14-s + 15-s + 16-s + 5.63·17-s + 18-s − 7.51·19-s + 20-s + 5.13·21-s + 2.22·22-s − 6.10·23-s + 24-s + 25-s − 6.60·26-s + 27-s + 5.13·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 1.94·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.669·11-s + 0.288·12-s − 1.83·13-s + 1.37·14-s + 0.258·15-s + 0.250·16-s + 1.36·17-s + 0.235·18-s − 1.72·19-s + 0.223·20-s + 1.12·21-s + 0.473·22-s − 1.27·23-s + 0.204·24-s + 0.200·25-s − 1.29·26-s + 0.192·27-s + 0.970·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.799593101\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.799593101\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 - 5.13T + 7T^{2} \) |
| 11 | \( 1 - 2.22T + 11T^{2} \) |
| 13 | \( 1 + 6.60T + 13T^{2} \) |
| 17 | \( 1 - 5.63T + 17T^{2} \) |
| 19 | \( 1 + 7.51T + 19T^{2} \) |
| 23 | \( 1 + 6.10T + 23T^{2} \) |
| 29 | \( 1 + 7.29T + 29T^{2} \) |
| 31 | \( 1 + 0.967T + 31T^{2} \) |
| 41 | \( 1 + 1.03T + 41T^{2} \) |
| 43 | \( 1 - 5.29T + 43T^{2} \) |
| 47 | \( 1 + 5.85T + 47T^{2} \) |
| 53 | \( 1 + 9.63T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 6.32T + 61T^{2} \) |
| 67 | \( 1 - 1.49T + 67T^{2} \) |
| 71 | \( 1 - 0.329T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 9.04T + 83T^{2} \) |
| 89 | \( 1 + 2.27T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−9.903600277335397607697018104168, −8.994595896041201859743668063080, −7.84641291040039046076858330195, −7.66756311934653492188336425223, −6.39276051051326307062073783110, −5.31704396473029676438462091860, −4.67205808356557498232867444452, −3.80199104308531425991935849404, −2.28095104184203708801932837653, −1.73599861733114192751296213316,
1.73599861733114192751296213316, 2.28095104184203708801932837653, 3.80199104308531425991935849404, 4.67205808356557498232867444452, 5.31704396473029676438462091860, 6.39276051051326307062073783110, 7.66756311934653492188336425223, 7.84641291040039046076858330195, 8.994595896041201859743668063080, 9.903600277335397607697018104168