L(s) = 1 | − 3-s + 5-s − 0.648·7-s + 9-s + 3.52·11-s + 1.35·13-s − 15-s + 6.87·17-s + 19-s + 0.648·21-s − 5.46·23-s + 25-s − 27-s + 5.52·29-s + 3.46·31-s − 3.52·33-s − 0.648·35-s + 0.0558·37-s − 1.35·39-s + 9.52·41-s − 7.69·43-s + 45-s + 1.46·47-s − 6.58·49-s − 6.87·51-s − 13.2·53-s + 3.52·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.244·7-s + 0.333·9-s + 1.06·11-s + 0.374·13-s − 0.258·15-s + 1.66·17-s + 0.229·19-s + 0.141·21-s − 1.14·23-s + 0.200·25-s − 0.192·27-s + 1.02·29-s + 0.622·31-s − 0.613·33-s − 0.109·35-s + 0.00917·37-s − 0.216·39-s + 1.48·41-s − 1.17·43-s + 0.149·45-s + 0.214·47-s − 0.940·49-s − 0.962·51-s − 1.81·53-s + 0.475·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.990697119\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.990697119\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 0.648T + 7T^{2} \) |
| 11 | \( 1 - 3.52T + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 - 6.87T + 17T^{2} \) |
| 23 | \( 1 + 5.46T + 23T^{2} \) |
| 29 | \( 1 - 5.52T + 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 - 0.0558T + 37T^{2} \) |
| 41 | \( 1 - 9.52T + 41T^{2} \) |
| 43 | \( 1 + 7.69T + 43T^{2} \) |
| 47 | \( 1 - 1.46T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 - 9.64T + 59T^{2} \) |
| 61 | \( 1 + 0.172T + 61T^{2} \) |
| 67 | \( 1 - 5.40T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 6.34T + 73T^{2} \) |
| 79 | \( 1 + 4.34T + 79T^{2} \) |
| 83 | \( 1 - 4.17T + 83T^{2} \) |
| 89 | \( 1 - 5.52T + 89T^{2} \) |
| 97 | \( 1 + 2.64T + 97T^{2} \) |
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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
−8.251223291590033612805251029074, −7.62374618072207524417828695642, −6.60227655585963548251272259010, −6.20883173791903007207221779654, −5.52700652470099430058778481720, −4.65789177339349409437211297211, −3.79289267317005557005522078376, −2.99961836171584115217742024377, −1.69758718302953965923419867408, −0.870164494203555565654132059981,
0.870164494203555565654132059981, 1.69758718302953965923419867408, 2.99961836171584115217742024377, 3.79289267317005557005522078376, 4.65789177339349409437211297211, 5.52700652470099430058778481720, 6.20883173791903007207221779654, 6.60227655585963548251272259010, 7.62374618072207524417828695642, 8.251223291590033612805251029074