L(s) = 1 | + 0.593·2-s + 1.93·3-s − 1.64·4-s − 5-s + 1.14·6-s + 1.06·7-s − 2.16·8-s + 0.725·9-s − 0.593·10-s + 0.196·11-s − 3.17·12-s + 5.28·13-s + 0.629·14-s − 1.93·15-s + 2.00·16-s + 0.704·17-s + 0.430·18-s + 1.64·20-s + 2.04·21-s + 0.116·22-s + 6.62·23-s − 4.18·24-s + 25-s + 3.13·26-s − 4.39·27-s − 1.74·28-s + 3.38·29-s + ⋯ |
L(s) = 1 | + 0.419·2-s + 1.11·3-s − 0.823·4-s − 0.447·5-s + 0.467·6-s + 0.400·7-s − 0.765·8-s + 0.241·9-s − 0.187·10-s + 0.0592·11-s − 0.917·12-s + 1.46·13-s + 0.168·14-s − 0.498·15-s + 0.502·16-s + 0.170·17-s + 0.101·18-s + 0.368·20-s + 0.446·21-s + 0.0248·22-s + 1.38·23-s − 0.853·24-s + 0.200·25-s + 0.615·26-s − 0.845·27-s − 0.330·28-s + 0.627·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.542522960\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.542522960\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 0.593T + 2T^{2} \) |
| 3 | \( 1 - 1.93T + 3T^{2} \) |
| 7 | \( 1 - 1.06T + 7T^{2} \) |
| 11 | \( 1 - 0.196T + 11T^{2} \) |
| 13 | \( 1 - 5.28T + 13T^{2} \) |
| 17 | \( 1 - 0.704T + 17T^{2} \) |
| 23 | \( 1 - 6.62T + 23T^{2} \) |
| 29 | \( 1 - 3.38T + 29T^{2} \) |
| 31 | \( 1 + 7.91T + 31T^{2} \) |
| 37 | \( 1 - 1.09T + 37T^{2} \) |
| 41 | \( 1 - 1.35T + 41T^{2} \) |
| 43 | \( 1 - 9.00T + 43T^{2} \) |
| 47 | \( 1 - 4.81T + 47T^{2} \) |
| 53 | \( 1 - 5.32T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 3.05T + 67T^{2} \) |
| 71 | \( 1 + 2.50T + 71T^{2} \) |
| 73 | \( 1 - 6.55T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 + 4.03T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 - 1.33T + 97T^{2} \) |
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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
−8.947978324034349676378707822876, −8.621766809414203961155863264331, −7.969319838556239768447354793774, −7.04559655182018588618791604672, −5.87172398596094112442932638077, −5.09860822225297586455630895807, −3.97548727411888383100205882615, −3.58751361162960603954872186055, −2.57959371274466554328873777931, −1.03440030517709971195313413570,
1.03440030517709971195313413570, 2.57959371274466554328873777931, 3.58751361162960603954872186055, 3.97548727411888383100205882615, 5.09860822225297586455630895807, 5.87172398596094112442932638077, 7.04559655182018588618791604672, 7.969319838556239768447354793774, 8.621766809414203961155863264331, 8.947978324034349676378707822876