| L(s) = 1 | − 3-s + 1.38·5-s − 4.85·7-s + 9-s + 3.61·11-s − 3.85·13-s − 1.38·15-s + 3.23·17-s + 7.23·19-s + 4.85·21-s − 4.47·23-s − 3.09·25-s − 27-s − 0.381·29-s + 8.47·31-s − 3.61·33-s − 6.70·35-s − 7.85·37-s + 3.85·39-s + 0.763·41-s − 5.85·43-s + 1.38·45-s + 2.61·47-s + 16.5·49-s − 3.23·51-s − 3.61·53-s + 5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.618·5-s − 1.83·7-s + 0.333·9-s + 1.09·11-s − 1.06·13-s − 0.356·15-s + 0.784·17-s + 1.66·19-s + 1.05·21-s − 0.932·23-s − 0.618·25-s − 0.192·27-s − 0.0709·29-s + 1.52·31-s − 0.629·33-s − 1.13·35-s − 1.29·37-s + 0.617·39-s + 0.119·41-s − 0.892·43-s + 0.206·45-s + 0.381·47-s + 2.36·49-s − 0.453·51-s − 0.496·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.255772018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.255772018\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 127 | \( 1 + T \) |
| good | 5 | \( 1 - 1.38T + 5T^{2} \) |
| 7 | \( 1 + 4.85T + 7T^{2} \) |
| 11 | \( 1 - 3.61T + 11T^{2} \) |
| 13 | \( 1 + 3.85T + 13T^{2} \) |
| 17 | \( 1 - 3.23T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 + 0.381T + 29T^{2} \) |
| 31 | \( 1 - 8.47T + 31T^{2} \) |
| 37 | \( 1 + 7.85T + 37T^{2} \) |
| 41 | \( 1 - 0.763T + 41T^{2} \) |
| 43 | \( 1 + 5.85T + 43T^{2} \) |
| 47 | \( 1 - 2.61T + 47T^{2} \) |
| 53 | \( 1 + 3.61T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 6.85T + 61T^{2} \) |
| 67 | \( 1 - 5.85T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 - 4.85T + 73T^{2} \) |
| 79 | \( 1 + 8.47T + 79T^{2} \) |
| 83 | \( 1 - 2.94T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 9.23T + 97T^{2} \) |
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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
−7.913040248014129435470836414447, −7.10385723971078092534817427402, −6.62409004370360894229763701589, −5.92931823684713172760012547419, −5.47147551624056798388110699566, −4.45555331914301342117825225764, −3.50067251317714533361782570379, −2.95307105859085809076600493045, −1.75347473805569931255762173184, −0.60237736770895278534462510465,
0.60237736770895278534462510465, 1.75347473805569931255762173184, 2.95307105859085809076600493045, 3.50067251317714533361782570379, 4.45555331914301342117825225764, 5.47147551624056798388110699566, 5.92931823684713172760012547419, 6.62409004370360894229763701589, 7.10385723971078092534817427402, 7.913040248014129435470836414447
