| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 10-s − 12-s + 15-s + 16-s − 20-s + 23-s + 24-s + 25-s + 27-s − 29-s − 30-s − 32-s + 40-s + 41-s + 43-s − 46-s + 2·47-s − 48-s − 50-s − 54-s + 58-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 10-s − 12-s + 15-s + 16-s − 20-s + 23-s + 24-s + 25-s + 27-s − 29-s − 30-s − 32-s + 40-s + 41-s + 43-s − 46-s + 2·47-s − 48-s − 50-s − 54-s + 58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3517709763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3517709763\) |
| \(L(1)\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( ( 1 - T )^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 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
−10.42354231321931317450699869928, −9.269874810140774012401872261697, −8.610891637354115887752619542777, −7.62027264082674202198978094764, −7.03056387031735253201866840097, −6.05881800475827923094127582578, −5.20071785912036012302023719718, −3.90623818502926466175496120981, −2.65504175226080796105475734693, −0.826539470459807720347956768792,
0.826539470459807720347956768792, 2.65504175226080796105475734693, 3.90623818502926466175496120981, 5.20071785912036012302023719718, 6.05881800475827923094127582578, 7.03056387031735253201866840097, 7.62027264082674202198978094764, 8.610891637354115887752619542777, 9.269874810140774012401872261697, 10.42354231321931317450699869928
