| L(s) = 1 | − 3-s + 5-s − 0.819·7-s + 9-s − 1.08·11-s + 13-s − 15-s + 1.18·17-s + 6.51·19-s + 0.819·21-s − 0.819·23-s + 25-s − 27-s + 8.51·29-s + 1.08·33-s − 0.819·35-s + 0.917·37-s − 39-s + 3.08·41-s − 8.41·43-s + 45-s − 6.32·49-s − 1.18·51-s − 7.49·53-s − 1.08·55-s − 6.51·57-s − 0.195·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.309·7-s + 0.333·9-s − 0.326·11-s + 0.277·13-s − 0.258·15-s + 0.286·17-s + 1.49·19-s + 0.178·21-s − 0.170·23-s + 0.200·25-s − 0.192·27-s + 1.58·29-s + 0.188·33-s − 0.138·35-s + 0.150·37-s − 0.160·39-s + 0.481·41-s − 1.28·43-s + 0.149·45-s − 0.904·49-s − 0.165·51-s − 1.02·53-s − 0.146·55-s − 0.862·57-s − 0.0255·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.731203058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.731203058\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 0.819T + 7T^{2} \) |
| 11 | \( 1 + 1.08T + 11T^{2} \) |
| 17 | \( 1 - 1.18T + 17T^{2} \) |
| 19 | \( 1 - 6.51T + 19T^{2} \) |
| 23 | \( 1 + 0.819T + 23T^{2} \) |
| 29 | \( 1 - 8.51T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 0.917T + 37T^{2} \) |
| 41 | \( 1 - 3.08T + 41T^{2} \) |
| 43 | \( 1 + 8.41T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 7.49T + 53T^{2} \) |
| 59 | \( 1 + 0.195T + 59T^{2} \) |
| 61 | \( 1 - 2.55T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 7.85T + 71T^{2} \) |
| 73 | \( 1 - 4.09T + 73T^{2} \) |
| 79 | \( 1 - 9.69T + 79T^{2} \) |
| 83 | \( 1 + 6.24T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−7.922244182048809529752328783259, −7.33921365061119411597108516915, −6.42381934089199490489878895528, −6.05164481953986986651741014949, −5.13280908439607691405708225910, −4.70241701608107679777518076221, −3.50165056530739153753232110827, −2.87617913367402781711432196526, −1.69030674720080461573724181021, −0.73129985731956819142392145770,
0.73129985731956819142392145770, 1.69030674720080461573724181021, 2.87617913367402781711432196526, 3.50165056530739153753232110827, 4.70241701608107679777518076221, 5.13280908439607691405708225910, 6.05164481953986986651741014949, 6.42381934089199490489878895528, 7.33921365061119411597108516915, 7.922244182048809529752328783259
