| L(s) = 1 | + 3-s − 5-s + 3.13·7-s + 9-s + 2.65·11-s + 0.651·13-s − 15-s + 4.48·17-s − 0.651·19-s + 3.13·21-s + 23-s + 25-s + 27-s + 2.48·29-s − 3.13·31-s + 2.65·33-s − 3.13·35-s − 6.10·37-s + 0.651·39-s + 0.820·41-s + 0.696·43-s − 45-s − 11.8·47-s + 2.83·49-s + 4.48·51-s + 9.45·53-s − 2.65·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.18·7-s + 0.333·9-s + 0.799·11-s + 0.180·13-s − 0.258·15-s + 1.08·17-s − 0.149·19-s + 0.684·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s + 0.461·29-s − 0.563·31-s + 0.461·33-s − 0.530·35-s − 1.00·37-s + 0.104·39-s + 0.128·41-s + 0.106·43-s − 0.149·45-s − 1.73·47-s + 0.404·49-s + 0.627·51-s + 1.29·53-s − 0.357·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.708239819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.708239819\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 3.13T + 7T^{2} \) |
| 11 | \( 1 - 2.65T + 11T^{2} \) |
| 13 | \( 1 - 0.651T + 13T^{2} \) |
| 17 | \( 1 - 4.48T + 17T^{2} \) |
| 19 | \( 1 + 0.651T + 19T^{2} \) |
| 29 | \( 1 - 2.48T + 29T^{2} \) |
| 31 | \( 1 + 3.13T + 31T^{2} \) |
| 37 | \( 1 + 6.10T + 37T^{2} \) |
| 41 | \( 1 - 0.820T + 41T^{2} \) |
| 43 | \( 1 - 0.696T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 - 9.45T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 - 2.65T + 61T^{2} \) |
| 67 | \( 1 + 6.10T + 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 - 6.31T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 5.51T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 5.93T + 97T^{2} \) |
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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
−8.521087825756044568372593781027, −8.272832137045385088506963782681, −7.38869181761207683737259504603, −6.75999155211541531730367684066, −5.60577069359647571916405529486, −4.83224004521179065913020157411, −3.96738912526065838987879982935, −3.26267788888737545650213533392, −2.00455269189953957288240224750, −1.09233158339460231921122061051,
1.09233158339460231921122061051, 2.00455269189953957288240224750, 3.26267788888737545650213533392, 3.96738912526065838987879982935, 4.83224004521179065913020157411, 5.60577069359647571916405529486, 6.75999155211541531730367684066, 7.38869181761207683737259504603, 8.272832137045385088506963782681, 8.521087825756044568372593781027
