| L(s) = 1 | − 2-s + 1.87·3-s + 4-s − 5-s − 1.87·6-s − 3.65·7-s − 8-s + 0.511·9-s + 10-s − 1.82·11-s + 1.87·12-s + 6.22·13-s + 3.65·14-s − 1.87·15-s + 16-s + 3.54·17-s − 0.511·18-s + 6.57·19-s − 20-s − 6.84·21-s + 1.82·22-s + 5.39·23-s − 1.87·24-s + 25-s − 6.22·26-s − 4.66·27-s − 3.65·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.08·3-s + 0.5·4-s − 0.447·5-s − 0.765·6-s − 1.37·7-s − 0.353·8-s + 0.170·9-s + 0.316·10-s − 0.551·11-s + 0.540·12-s + 1.72·13-s + 0.975·14-s − 0.483·15-s + 0.250·16-s + 0.859·17-s − 0.120·18-s + 1.50·19-s − 0.223·20-s − 1.49·21-s + 0.389·22-s + 1.12·23-s − 0.382·24-s + 0.200·25-s − 1.22·26-s − 0.897·27-s − 0.689·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.731994121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.731994121\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 - 1.87T + 3T^{2} \) |
| 7 | \( 1 + 3.65T + 7T^{2} \) |
| 11 | \( 1 + 1.82T + 11T^{2} \) |
| 13 | \( 1 - 6.22T + 13T^{2} \) |
| 17 | \( 1 - 3.54T + 17T^{2} \) |
| 19 | \( 1 - 6.57T + 19T^{2} \) |
| 23 | \( 1 - 5.39T + 23T^{2} \) |
| 29 | \( 1 - 8.94T + 29T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 8.03T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 - 0.821T + 47T^{2} \) |
| 53 | \( 1 + 4.11T + 53T^{2} \) |
| 59 | \( 1 + 3.42T + 59T^{2} \) |
| 61 | \( 1 + 9.38T + 61T^{2} \) |
| 67 | \( 1 + 0.182T + 67T^{2} \) |
| 71 | \( 1 - 1.88T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 + 5.06T + 79T^{2} \) |
| 83 | \( 1 + 2.69T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 5.70T + 97T^{2} \) |
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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
−7.87691523417133398989422957948, −7.13605911720767903007120960447, −6.56785982632239831602485428067, −5.80346634787438051903023632591, −4.99541167828573251309624565512, −3.63293526976788889678710446069, −3.15063477002250620772954931636, −3.03684132912234184012988947210, −1.62002881471712448260047047747, −0.67686425405655737683174966119,
0.67686425405655737683174966119, 1.62002881471712448260047047747, 3.03684132912234184012988947210, 3.15063477002250620772954931636, 3.63293526976788889678710446069, 4.99541167828573251309624565512, 5.80346634787438051903023632591, 6.56785982632239831602485428067, 7.13605911720767903007120960447, 7.87691523417133398989422957948
