| L(s) = 1 | − 4·5-s + 4·13-s + 2·17-s + 11·25-s − 4·29-s − 12·37-s + 10·41-s − 7·49-s − 4·53-s − 12·61-s − 16·65-s − 6·73-s − 8·85-s − 10·89-s − 18·97-s − 20·101-s + 20·109-s + 14·113-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.10·13-s + 0.485·17-s + 11/5·25-s − 0.742·29-s − 1.97·37-s + 1.56·41-s − 49-s − 0.549·53-s − 1.53·61-s − 1.98·65-s − 0.702·73-s − 0.867·85-s − 1.05·89-s − 1.82·97-s − 1.99·101-s + 1.91·109-s + 1.31·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 18 T + p T^{2} \) |
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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
−8.518580926433601982373809526294, −7.83885085572920869613116207995, −7.27777410584902448752390598516, −6.39063738730332703469978407006, −5.39010496228517083389897023205, −4.38564257449405141873091168941, −3.71637470346215322038560926740, −3.04380780885898707635692948137, −1.36924256495576692234243060544, 0,
1.36924256495576692234243060544, 3.04380780885898707635692948137, 3.71637470346215322038560926740, 4.38564257449405141873091168941, 5.39010496228517083389897023205, 6.39063738730332703469978407006, 7.27777410584902448752390598516, 7.83885085572920869613116207995, 8.518580926433601982373809526294
