| L(s) = 1 | + 9.05e3·5-s − 1.68e4·7-s + 1.01e6·11-s − 2.12e6·13-s − 5.48e6·17-s + 2.07e7·19-s − 5.46e7·23-s + 3.31e7·25-s − 1.42e8·29-s + 1.72e8·31-s − 1.52e8·35-s − 6.75e8·37-s + 9.09e7·41-s − 4.27e8·43-s + 2.38e8·47-s + 2.82e8·49-s − 1.72e9·53-s + 9.16e9·55-s − 5.53e9·59-s − 3.08e9·61-s − 1.92e10·65-s − 9.55e9·67-s − 1.00e10·71-s − 2.52e10·73-s − 1.70e10·77-s + 2.70e10·79-s + 6.76e10·83-s + ⋯ |
| L(s) = 1 | + 1.29·5-s − 0.377·7-s + 1.89·11-s − 1.58·13-s − 0.937·17-s + 1.92·19-s − 1.77·23-s + 0.679·25-s − 1.29·29-s + 1.08·31-s − 0.489·35-s − 1.60·37-s + 0.122·41-s − 0.443·43-s + 0.151·47-s + 0.142·49-s − 0.567·53-s + 2.45·55-s − 1.00·59-s − 0.468·61-s − 2.05·65-s − 0.864·67-s − 0.662·71-s − 1.42·73-s − 0.716·77-s + 0.988·79-s + 1.88·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 1.68e4T \) |
| good | 5 | \( 1 - 9.05e3T + 4.88e7T^{2} \) |
| 11 | \( 1 - 1.01e6T + 2.85e11T^{2} \) |
| 13 | \( 1 + 2.12e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 5.48e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 2.07e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 5.46e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.42e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.72e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 6.75e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 9.09e7T + 5.50e17T^{2} \) |
| 43 | \( 1 + 4.27e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.38e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.72e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 5.53e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 3.08e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 9.55e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.00e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.52e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.70e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 6.76e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 1.58e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 5.58e10T + 7.15e21T^{2} \) |
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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
−9.537805342822995867680036019315, −9.124113216475991198256955096376, −7.51274810518000702381568490663, −6.57548188510520335885819835335, −5.79730639288564638603743735622, −4.68283152540981081517817398216, −3.44483691533602349418076044154, −2.17585416847733003985819703915, −1.42250601299090149642965844452, 0,
1.42250601299090149642965844452, 2.17585416847733003985819703915, 3.44483691533602349418076044154, 4.68283152540981081517817398216, 5.79730639288564638603743735622, 6.57548188510520335885819835335, 7.51274810518000702381568490663, 9.124113216475991198256955096376, 9.537805342822995867680036019315
