| L(s) = 1 | + 10·5-s − 4·7-s + 20·11-s − 70·13-s − 90·17-s − 140·19-s + 192·23-s − 25·25-s − 134·29-s + 100·31-s − 40·35-s + 170·37-s + 110·41-s − 532·43-s + 56·47-s − 327·49-s − 430·53-s + 200·55-s − 20·59-s − 270·61-s − 700·65-s + 524·67-s + 80·71-s + 330·73-s − 80·77-s + 1.06e3·79-s − 1.18e3·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.215·7-s + 0.548·11-s − 1.49·13-s − 1.28·17-s − 1.69·19-s + 1.74·23-s − 1/5·25-s − 0.858·29-s + 0.579·31-s − 0.193·35-s + 0.755·37-s + 0.419·41-s − 1.88·43-s + 0.173·47-s − 0.953·49-s − 1.11·53-s + 0.490·55-s − 0.0441·59-s − 0.566·61-s − 1.33·65-s + 0.955·67-s + 0.133·71-s + 0.529·73-s − 0.118·77-s + 1.50·79-s − 1.57·83-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 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 7 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 70 T + p^{3} T^{2} \) |
| 17 | \( 1 + 90 T + p^{3} T^{2} \) |
| 19 | \( 1 + 140 T + p^{3} T^{2} \) |
| 23 | \( 1 - 192 T + p^{3} T^{2} \) |
| 29 | \( 1 + 134 T + p^{3} T^{2} \) |
| 31 | \( 1 - 100 T + p^{3} T^{2} \) |
| 37 | \( 1 - 170 T + p^{3} T^{2} \) |
| 41 | \( 1 - 110 T + p^{3} T^{2} \) |
| 43 | \( 1 + 532 T + p^{3} T^{2} \) |
| 47 | \( 1 - 56 T + p^{3} T^{2} \) |
| 53 | \( 1 + 430 T + p^{3} T^{2} \) |
| 59 | \( 1 + 20 T + p^{3} T^{2} \) |
| 61 | \( 1 + 270 T + p^{3} T^{2} \) |
| 67 | \( 1 - 524 T + p^{3} T^{2} \) |
| 71 | \( 1 - 80 T + p^{3} T^{2} \) |
| 73 | \( 1 - 330 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1060 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1188 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1274 T + p^{3} T^{2} \) |
| 97 | \( 1 + 590 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.694397508797212442745035212872, −9.245377962400217435059977358078, −8.208332681649884755425609288332, −6.89199337660583853317715284780, −6.41716780902006901716200848148, −5.16741838372930162984051879029, −4.30632769635815019718880272138, −2.74634064417202034913997095541, −1.81110578588999287003461155390, 0,
1.81110578588999287003461155390, 2.74634064417202034913997095541, 4.30632769635815019718880272138, 5.16741838372930162984051879029, 6.41716780902006901716200848148, 6.89199337660583853317715284780, 8.208332681649884755425609288332, 9.245377962400217435059977358078, 9.694397508797212442745035212872
