| L(s) = 1 | − 3·3-s − 6·5-s + 16·7-s + 9·9-s + 12·11-s − 38·13-s + 18·15-s − 126·17-s + 20·19-s − 48·21-s − 168·23-s − 89·25-s − 27·27-s − 30·29-s + 88·31-s − 36·33-s − 96·35-s − 254·37-s + 114·39-s + 42·41-s − 52·43-s − 54·45-s + 96·47-s − 87·49-s + 378·51-s − 198·53-s − 72·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.536·5-s + 0.863·7-s + 1/3·9-s + 0.328·11-s − 0.810·13-s + 0.309·15-s − 1.79·17-s + 0.241·19-s − 0.498·21-s − 1.52·23-s − 0.711·25-s − 0.192·27-s − 0.192·29-s + 0.509·31-s − 0.189·33-s − 0.463·35-s − 1.12·37-s + 0.468·39-s + 0.159·41-s − 0.184·43-s − 0.178·45-s + 0.297·47-s − 0.253·49-s + 1.03·51-s − 0.513·53-s − 0.176·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| good | 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 126 T + p^{3} T^{2} \) |
| 19 | \( 1 - 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + 168 T + p^{3} T^{2} \) |
| 29 | \( 1 + 30 T + p^{3} T^{2} \) |
| 31 | \( 1 - 88 T + p^{3} T^{2} \) |
| 37 | \( 1 + 254 T + p^{3} T^{2} \) |
| 41 | \( 1 - 42 T + p^{3} T^{2} \) |
| 43 | \( 1 + 52 T + p^{3} T^{2} \) |
| 47 | \( 1 - 96 T + p^{3} T^{2} \) |
| 53 | \( 1 + 198 T + p^{3} T^{2} \) |
| 59 | \( 1 + 660 T + p^{3} T^{2} \) |
| 61 | \( 1 - 538 T + p^{3} T^{2} \) |
| 67 | \( 1 - 884 T + p^{3} T^{2} \) |
| 71 | \( 1 + 792 T + p^{3} T^{2} \) |
| 73 | \( 1 - 218 T + p^{3} T^{2} \) |
| 79 | \( 1 - 520 T + p^{3} T^{2} \) |
| 83 | \( 1 + 492 T + p^{3} T^{2} \) |
| 89 | \( 1 - 810 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1154 T + p^{3} T^{2} \) |
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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
−11.63257493869608171188933507304, −10.83523076823632669724894007274, −9.687702124874222536838500009932, −8.437900006570885184688670562765, −7.46976092026210783734304435064, −6.35012562228816624310614608230, −4.97753441386872764896400612379, −4.05994039875513833613158376426, −1.99557943019966377695262793946, 0,
1.99557943019966377695262793946, 4.05994039875513833613158376426, 4.97753441386872764896400612379, 6.35012562228816624310614608230, 7.46976092026210783734304435064, 8.437900006570885184688670562765, 9.687702124874222536838500009932, 10.83523076823632669724894007274, 11.63257493869608171188933507304
