| L(s) = 1 | + 625·5-s + 9.62e3·7-s − 5.56e4·11-s + 1.69e5·13-s − 2.07e5·17-s + 8.02e5·19-s + 1.24e6·23-s + 3.90e5·25-s + 4.28e6·29-s − 3.58e6·31-s + 6.01e6·35-s − 2.89e6·37-s − 2.51e7·41-s − 2.00e7·43-s − 3.73e7·47-s + 5.22e7·49-s + 2.55e7·53-s − 3.47e7·55-s + 9.96e7·59-s + 2.00e8·61-s + 1.06e8·65-s − 8.09e7·67-s + 4.31e7·71-s − 3.40e8·73-s − 5.35e8·77-s + 2.81e8·79-s + 6.01e8·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s − 1.14·11-s + 1.64·13-s − 0.602·17-s + 1.41·19-s + 0.925·23-s + 0.200·25-s + 1.12·29-s − 0.697·31-s + 0.677·35-s − 0.254·37-s − 1.39·41-s − 0.893·43-s − 1.11·47-s + 1.29·49-s + 0.444·53-s − 0.512·55-s + 1.07·59-s + 1.85·61-s + 0.737·65-s − 0.491·67-s + 0.201·71-s − 1.40·73-s − 1.73·77-s + 0.811·79-s + 1.39·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.222302375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.222302375\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 625T \) |
| good | 7 | \( 1 - 9.62e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.56e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.69e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.07e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.02e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.24e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.28e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.58e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.89e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.51e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.00e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.73e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.55e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.96e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.00e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.09e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 4.31e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.40e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.81e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.01e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.39e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 4.23e8T + 7.60e17T^{2} \) |
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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
−11.04115131705730604604559046270, −10.16305367744356204892571373291, −8.734107315910264722179841313724, −8.130125615669988024223430332740, −6.91114814128166393906382219652, −5.50791507519377967773874353063, −4.83728718943836464606876265653, −3.28962010403200739373232580122, −1.90154876739565485109956218788, −0.945763263855134365869180153532,
0.945763263855134365869180153532, 1.90154876739565485109956218788, 3.28962010403200739373232580122, 4.83728718943836464606876265653, 5.50791507519377967773874353063, 6.91114814128166393906382219652, 8.130125615669988024223430332740, 8.734107315910264722179841313724, 10.16305367744356204892571373291, 11.04115131705730604604559046270
