| L(s) = 1 | + 2·3-s + 2·5-s + 7-s + 9-s − 2·11-s − 4·13-s + 4·15-s + 2·17-s + 2·21-s − 25-s − 4·27-s + 2·29-s − 4·33-s + 2·35-s − 4·37-s − 8·39-s + 6·41-s − 2·43-s + 2·45-s − 4·47-s + 49-s + 4·51-s − 4·55-s + 2·59-s − 10·61-s + 63-s − 8·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s + 1.03·15-s + 0.485·17-s + 0.436·21-s − 1/5·25-s − 0.769·27-s + 0.371·29-s − 0.696·33-s + 0.338·35-s − 0.657·37-s − 1.28·39-s + 0.937·41-s − 0.304·43-s + 0.298·45-s − 0.583·47-s + 1/7·49-s + 0.560·51-s − 0.539·55-s + 0.260·59-s − 1.28·61-s + 0.125·63-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.629097423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.629097423\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.91473695366310, −12.61049552879091, −12.05305427277908, −11.54376742674906, −10.97964156129171, −10.41863444744263, −9.921597808938928, −9.703850450616014, −9.186613919749506, −8.665498871491103, −8.254314213266969, −7.724122148658278, −7.394772655410814, −6.882989311931898, −6.041022267369215, −5.782276275834478, −5.106072678539352, −4.734771836415736, −4.087205354969279, −3.337162777906916, −2.950986454201428, −2.362278493966043, −1.996411943043085, −1.425528753002578, −0.4453638408456986,
0.4453638408456986, 1.425528753002578, 1.996411943043085, 2.362278493966043, 2.950986454201428, 3.337162777906916, 4.087205354969279, 4.734771836415736, 5.106072678539352, 5.782276275834478, 6.041022267369215, 6.882989311931898, 7.394772655410814, 7.724122148658278, 8.254314213266969, 8.665498871491103, 9.186613919749506, 9.703850450616014, 9.921597808938928, 10.41863444744263, 10.97964156129171, 11.54376742674906, 12.05305427277908, 12.61049552879091, 12.91473695366310
