| L(s) = 1 | − 3-s − 5-s − 7-s − 2·9-s + 3·11-s − 5·13-s + 15-s + 3·17-s + 21-s − 23-s + 25-s + 5·27-s − 3·29-s − 6·31-s − 3·33-s + 35-s − 4·37-s + 5·39-s − 2·41-s + 8·43-s + 2·45-s + 47-s + 49-s − 3·51-s + 4·53-s − 3·55-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.904·11-s − 1.38·13-s + 0.258·15-s + 0.727·17-s + 0.218·21-s − 0.208·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s − 1.07·31-s − 0.522·33-s + 0.169·35-s − 0.657·37-s + 0.800·39-s − 0.312·41-s + 1.21·43-s + 0.298·45-s + 0.145·47-s + 1/7·49-s − 0.420·51-s + 0.549·53-s − 0.404·55-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−14.63371764469881, −14.35646253115656, −13.89494896088185, −13.01112862840855, −12.59612517271542, −12.01545064396904, −11.83342823360974, −11.26587311552031, −10.62985643691612, −10.17885808917007, −9.516198363957380, −9.068008143759650, −8.577983996983087, −7.721508859428035, −7.394989174706626, −6.782066757857465, −6.228732470028773, −5.502969993556985, −5.255038535651152, −4.404982394688798, −3.836138720596201, −3.196469783332188, −2.538045186578772, −1.714340672844668, −0.7239921949775115, 0,
0.7239921949775115, 1.714340672844668, 2.538045186578772, 3.196469783332188, 3.836138720596201, 4.404982394688798, 5.255038535651152, 5.502969993556985, 6.228732470028773, 6.782066757857465, 7.394989174706626, 7.721508859428035, 8.577983996983087, 9.068008143759650, 9.516198363957380, 10.17885808917007, 10.62985643691612, 11.26587311552031, 11.83342823360974, 12.01545064396904, 12.59612517271542, 13.01112862840855, 13.89494896088185, 14.35646253115656, 14.63371764469881
