L(s) = 1 | − 0.618·3-s + 2.85·5-s + 1.23·7-s − 2.61·9-s + 3.61·11-s − 3.85·13-s − 1.76·15-s + 4.47·17-s + 4.47·19-s − 0.763·21-s − 3.85·23-s + 3.14·25-s + 3.47·27-s − 6.32·29-s + 9.61·31-s − 2.23·33-s + 3.52·35-s + 37-s + 2.38·39-s + 7.38·41-s + 0.763·43-s − 7.47·45-s + 3.23·47-s − 5.47·49-s − 2.76·51-s + 8.47·53-s + 10.3·55-s + ⋯ |
L(s) = 1 | − 0.356·3-s + 1.27·5-s + 0.467·7-s − 0.872·9-s + 1.09·11-s − 1.06·13-s − 0.455·15-s + 1.08·17-s + 1.02·19-s − 0.166·21-s − 0.803·23-s + 0.629·25-s + 0.668·27-s − 1.17·29-s + 1.72·31-s − 0.389·33-s + 0.596·35-s + 0.164·37-s + 0.381·39-s + 1.15·41-s + 0.116·43-s − 1.11·45-s + 0.472·47-s − 0.781·49-s − 0.387·51-s + 1.16·53-s + 1.39·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.150803541\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.150803541\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 37 | \( 1 - T \) |
good | 3 | \( 1 + 0.618T + 3T^{2} \) |
| 5 | \( 1 - 2.85T + 5T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 - 3.61T + 11T^{2} \) |
| 13 | \( 1 + 3.85T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 + 3.85T + 23T^{2} \) |
| 29 | \( 1 + 6.32T + 29T^{2} \) |
| 31 | \( 1 - 9.61T + 31T^{2} \) |
| 41 | \( 1 - 7.38T + 41T^{2} \) |
| 43 | \( 1 - 0.763T + 43T^{2} \) |
| 47 | \( 1 - 3.23T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 9.23T + 59T^{2} \) |
| 61 | \( 1 + 8.38T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + 4.09T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 - 5.52T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
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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
−9.211914512529095991765098812868, −8.202685620827620245425193388616, −7.43232716833984782428259159584, −6.46655843021428799101481980599, −5.71969753473980120843999056596, −5.33321532635402224250754161904, −4.26790407297697050878583959048, −3.03452105728171245830533755360, −2.10835005658084739767123437497, −1.00943523764541832593960189408,
1.00943523764541832593960189408, 2.10835005658084739767123437497, 3.03452105728171245830533755360, 4.26790407297697050878583959048, 5.33321532635402224250754161904, 5.71969753473980120843999056596, 6.46655843021428799101481980599, 7.43232716833984782428259159584, 8.202685620827620245425193388616, 9.211914512529095991765098812868