| L(s) = 1 | − 2·5-s + 7-s − 6·11-s − 6·13-s + 2·17-s + 4·19-s − 2·23-s − 25-s − 8·29-s + 4·31-s − 2·35-s − 6·37-s − 10·41-s − 4·43-s + 4·47-s + 49-s + 4·53-s + 12·55-s + 12·59-s − 2·61-s + 12·65-s + 12·67-s − 6·71-s − 2·73-s − 6·77-s − 8·79-s − 4·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 1.80·11-s − 1.66·13-s + 0.485·17-s + 0.917·19-s − 0.417·23-s − 1/5·25-s − 1.48·29-s + 0.718·31-s − 0.338·35-s − 0.986·37-s − 1.56·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.549·53-s + 1.61·55-s + 1.56·59-s − 0.256·61-s + 1.48·65-s + 1.46·67-s − 0.712·71-s − 0.234·73-s − 0.683·77-s − 0.900·79-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−10.35008322056724847315356615191, −9.836414938364711016506553782738, −8.443250331630403097115425862777, −7.65040352024265423959330412357, −7.24055876776084908676498360550, −5.45714738809688967843005370349, −4.90573494789049213587069738538, −3.52674311972011206774885342352, −2.31566940950309691908315799446, 0,
2.31566940950309691908315799446, 3.52674311972011206774885342352, 4.90573494789049213587069738538, 5.45714738809688967843005370349, 7.24055876776084908676498360550, 7.65040352024265423959330412357, 8.443250331630403097115425862777, 9.836414938364711016506553782738, 10.35008322056724847315356615191
