| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s + 4·11-s + 15-s + 17-s − 4·19-s + 2·21-s + 25-s − 27-s − 8·31-s − 4·33-s + 2·35-s − 2·37-s + 2·41-s − 6·43-s − 45-s + 12·47-s − 3·49-s − 51-s + 12·53-s − 4·55-s + 4·57-s − 2·63-s − 12·67-s + 12·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.258·15-s + 0.242·17-s − 0.917·19-s + 0.436·21-s + 1/5·25-s − 0.192·27-s − 1.43·31-s − 0.696·33-s + 0.338·35-s − 0.328·37-s + 0.312·41-s − 0.914·43-s − 0.149·45-s + 1.75·47-s − 3/7·49-s − 0.140·51-s + 1.64·53-s − 0.539·55-s + 0.529·57-s − 0.251·63-s − 1.46·67-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.72513500745163, −12.18879758531503, −12.06162633918164, −11.35206916350993, −11.11947188568205, −10.46496238044488, −10.22877904006229, −9.529778019139476, −9.190827202199780, −8.743859468749772, −8.301584292526134, −7.521508109110837, −7.220486175329357, −6.726426149930472, −6.273439568168086, −5.922333200230290, −5.294838342472979, −4.784873423685039, −4.148981928103873, −3.649674144140812, −3.528215259555639, −2.549586259226125, −2.026150186505044, −1.297852308807397, −0.6671417197582260, 0,
0.6671417197582260, 1.297852308807397, 2.026150186505044, 2.549586259226125, 3.528215259555639, 3.649674144140812, 4.148981928103873, 4.784873423685039, 5.294838342472979, 5.922333200230290, 6.273439568168086, 6.726426149930472, 7.220486175329357, 7.521508109110837, 8.301584292526134, 8.743859468749772, 9.190827202199780, 9.529778019139476, 10.22877904006229, 10.46496238044488, 11.11947188568205, 11.35206916350993, 12.06162633918164, 12.18879758531503, 12.72513500745163
