| L(s) = 1 | − 17·7-s + 89·13-s − 107·19-s − 125·25-s − 308·31-s − 433·37-s + 520·43-s − 54·49-s − 901·61-s − 1.00e3·67-s − 271·73-s − 503·79-s − 1.51e3·91-s + 1.85e3·97-s + 19·103-s − 646·109-s + ⋯ |
| L(s) = 1 | − 0.917·7-s + 1.89·13-s − 1.29·19-s − 25-s − 1.78·31-s − 1.92·37-s + 1.84·43-s − 0.157·49-s − 1.89·61-s − 1.83·67-s − 0.434·73-s − 0.716·79-s − 1.74·91-s + 1.93·97-s + 0.0181·103-s − 0.567·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 + 17 T + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 89 T + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 + 107 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 + 308 T + p^{3} T^{2} \) |
| 37 | \( 1 + 433 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 - 520 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 + 901 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1007 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 + 271 T + p^{3} T^{2} \) |
| 79 | \( 1 + 503 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 - 1853 T + p^{3} T^{2} \) |
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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
−10.47410496416926696399755571037, −9.214098552570282057479399903102, −8.662978880402692178115805578285, −7.46137311510740541549546176640, −6.35137811467381880015457517853, −5.75906467816420832349565254987, −4.13258105070953970658220289272, −3.31977084184142865726692630528, −1.72160732391604166166865569314, 0,
1.72160732391604166166865569314, 3.31977084184142865726692630528, 4.13258105070953970658220289272, 5.75906467816420832349565254987, 6.35137811467381880015457517853, 7.46137311510740541549546176640, 8.662978880402692178115805578285, 9.214098552570282057479399903102, 10.47410496416926696399755571037
