| L(s) = 1 | + 2-s + 0.347·3-s + 4-s + 0.347·6-s − 1.87·7-s + 8-s − 0.879·9-s + 0.347·12-s + 1.53·13-s − 1.87·14-s + 16-s + 1.53·17-s − 0.879·18-s + 19-s − 0.652·21-s + 0.347·23-s + 0.347·24-s + 1.53·26-s − 0.652·27-s − 1.87·28-s + 1.53·29-s + 32-s + 1.53·34-s − 0.879·36-s − 37-s + 38-s + 0.532·39-s + ⋯ |
| L(s) = 1 | + 2-s + 0.347·3-s + 4-s + 0.347·6-s − 1.87·7-s + 8-s − 0.879·9-s + 0.347·12-s + 1.53·13-s − 1.87·14-s + 16-s + 1.53·17-s − 0.879·18-s + 19-s − 0.652·21-s + 0.347·23-s + 0.347·24-s + 1.53·26-s − 0.652·27-s − 1.87·28-s + 1.53·29-s + 32-s + 1.53·34-s − 0.879·36-s − 37-s + 38-s + 0.532·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.480289576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.480289576\) |
| \(L(1)\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.347T + T^{2} \) |
| 7 | \( 1 + 1.87T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.53T + T^{2} \) |
| 17 | \( 1 - 1.53T + T^{2} \) |
| 23 | \( 1 - 0.347T + T^{2} \) |
| 29 | \( 1 - 1.53T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + 1.87T + T^{2} \) |
| 59 | \( 1 - 0.347T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.87T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.347T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.615767858306597453945641804078, −7.85810725251667225240945586182, −6.96872483613053770117503391512, −6.20992304248875064512757145322, −5.88004712399777255206150725396, −4.99621449367105578287382345469, −3.66230199216777753740237343710, −3.27841165432932801604987387238, −2.83834727189313551016203009367, −1.24394717335128879486726389519,
1.24394717335128879486726389519, 2.83834727189313551016203009367, 3.27841165432932801604987387238, 3.66230199216777753740237343710, 4.99621449367105578287382345469, 5.88004712399777255206150725396, 6.20992304248875064512757145322, 6.96872483613053770117503391512, 7.85810725251667225240945586182, 8.615767858306597453945641804078
