| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s − 12-s − 13-s + 14-s + 16-s + 17-s − 19-s − 21-s + 23-s − 24-s − 26-s + 27-s + 28-s + 29-s + 32-s + 34-s + 2·37-s − 38-s + 39-s − 42-s + 46-s − 2·47-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s − 12-s − 13-s + 14-s + 16-s + 17-s − 19-s − 21-s + 23-s − 24-s − 26-s + 27-s + 28-s + 29-s + 32-s + 34-s + 2·37-s − 38-s + 39-s − 42-s + 46-s − 2·47-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.977651916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.977651916\) |
| \(L(1)\) |
|
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 | \( 1 + T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 + T )^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−8.344594813413036440170882909148, −7.84043408226078188746646011261, −6.94179131692836676328005507316, −6.28332084006563039557344314495, −5.57317961866562438810972265169, −4.75866632764358507382137257146, −4.62228818441908021070726446133, −3.24022384858830867970215008769, −2.36626740426791667930896507752, −1.17694312738159186309599882069,
1.17694312738159186309599882069, 2.36626740426791667930896507752, 3.24022384858830867970215008769, 4.62228818441908021070726446133, 4.75866632764358507382137257146, 5.57317961866562438810972265169, 6.28332084006563039557344314495, 6.94179131692836676328005507316, 7.84043408226078188746646011261, 8.344594813413036440170882909148
