| L(s) = 1 | + 4·5-s − 18·13-s − 104·17-s − 109·25-s + 284·29-s − 214·37-s − 472·41-s − 343·49-s + 572·53-s − 830·61-s − 72·65-s − 1.09e3·73-s − 416·85-s + 176·89-s − 594·97-s − 1.94e3·101-s − 1.74e3·109-s + 1.32e3·113-s + ⋯ |
| L(s) = 1 | + 0.357·5-s − 0.384·13-s − 1.48·17-s − 0.871·25-s + 1.81·29-s − 0.950·37-s − 1.79·41-s − 49-s + 1.48·53-s − 1.74·61-s − 0.137·65-s − 1.76·73-s − 0.530·85-s + 0.209·89-s − 0.621·97-s − 1.91·101-s − 1.53·109-s + 1.10·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 + 18 T + p^{3} T^{2} \) |
| 17 | \( 1 + 104 T + p^{3} T^{2} \) |
| 19 | \( 1 + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 - 284 T + p^{3} T^{2} \) |
| 31 | \( 1 + p^{3} T^{2} \) |
| 37 | \( 1 + 214 T + p^{3} T^{2} \) |
| 41 | \( 1 + 472 T + p^{3} T^{2} \) |
| 43 | \( 1 + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 - 572 T + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 + 830 T + p^{3} T^{2} \) |
| 67 | \( 1 + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 + 1098 T + p^{3} T^{2} \) |
| 79 | \( 1 + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 - 176 T + p^{3} T^{2} \) |
| 97 | \( 1 + 594 T + p^{3} T^{2} \) |
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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
−9.961298787262456769621518672452, −8.969302674038261400762958269182, −8.239074330365357145982852365237, −7.03811931015362881887821006700, −6.32814677262040118027637646148, −5.16894222994826421885905171500, −4.24813172295872249015471040529, −2.86984184630707362193903160492, −1.71918663622732346875492628809, 0,
1.71918663622732346875492628809, 2.86984184630707362193903160492, 4.24813172295872249015471040529, 5.16894222994826421885905171500, 6.32814677262040118027637646148, 7.03811931015362881887821006700, 8.239074330365357145982852365237, 8.969302674038261400762958269182, 9.961298787262456769621518672452
