| L(s) = 1 | − 0.144·2-s + 3-s − 1.97·4-s + 1.55·5-s − 0.144·6-s − 1.52·7-s + 0.574·8-s + 9-s − 0.224·10-s − 0.767·11-s − 1.97·12-s − 2.61·13-s + 0.220·14-s + 1.55·15-s + 3.87·16-s − 3.51·17-s − 0.144·18-s + 5.24·19-s − 3.07·20-s − 1.52·21-s + 0.110·22-s + 8.49·23-s + 0.574·24-s − 2.58·25-s + 0.377·26-s + 27-s + 3.02·28-s + ⋯ |
| L(s) = 1 | − 0.102·2-s + 0.577·3-s − 0.989·4-s + 0.695·5-s − 0.0589·6-s − 0.577·7-s + 0.203·8-s + 0.333·9-s − 0.0709·10-s − 0.231·11-s − 0.571·12-s − 0.726·13-s + 0.0589·14-s + 0.401·15-s + 0.968·16-s − 0.852·17-s − 0.0340·18-s + 1.20·19-s − 0.687·20-s − 0.333·21-s + 0.0236·22-s + 1.77·23-s + 0.117·24-s − 0.516·25-s + 0.0741·26-s + 0.192·27-s + 0.571·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.669572283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.669572283\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 0.144T + 2T^{2} \) |
| 5 | \( 1 - 1.55T + 5T^{2} \) |
| 7 | \( 1 + 1.52T + 7T^{2} \) |
| 11 | \( 1 + 0.767T + 11T^{2} \) |
| 13 | \( 1 + 2.61T + 13T^{2} \) |
| 17 | \( 1 + 3.51T + 17T^{2} \) |
| 19 | \( 1 - 5.24T + 19T^{2} \) |
| 23 | \( 1 - 8.49T + 23T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 + 3.00T + 37T^{2} \) |
| 41 | \( 1 - 5.79T + 41T^{2} \) |
| 43 | \( 1 - 4.29T + 43T^{2} \) |
| 47 | \( 1 - 1.77T + 47T^{2} \) |
| 53 | \( 1 - 6.77T + 53T^{2} \) |
| 59 | \( 1 - 14.9T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 7.29T + 67T^{2} \) |
| 71 | \( 1 + 4.81T + 71T^{2} \) |
| 73 | \( 1 - 9.19T + 73T^{2} \) |
| 79 | \( 1 - 0.545T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 1.77T + 97T^{2} \) |
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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
−9.210375907212062813786111151244, −8.301741840946682430188224384520, −7.41659548121123750374721641780, −6.76541088176346544800293683128, −5.57395986418026934343744725035, −5.05689275118751751026067913563, −4.05829933450750100228103494494, −3.14610368748903989360520842296, −2.24123475052888654306924849141, −0.818623466191123082196960760735,
0.818623466191123082196960760735, 2.24123475052888654306924849141, 3.14610368748903989360520842296, 4.05829933450750100228103494494, 5.05689275118751751026067913563, 5.57395986418026934343744725035, 6.76541088176346544800293683128, 7.41659548121123750374721641780, 8.301741840946682430188224384520, 9.210375907212062813786111151244
