| L(s) = 1 | − 2-s + 3-s + 5-s − 6-s + 8-s + 9-s − 10-s − 11-s − 13-s + 15-s − 16-s − 18-s + 22-s − 23-s + 24-s + 25-s + 26-s + 27-s + 2·29-s − 30-s + 2·31-s − 33-s + 2·37-s − 39-s + 40-s − 41-s + 45-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 5-s − 6-s + 8-s + 9-s − 10-s − 11-s − 13-s + 15-s − 16-s − 18-s + 22-s − 23-s + 24-s + 25-s + 26-s + 27-s + 2·29-s − 30-s + 2·31-s − 33-s + 2·37-s − 39-s + 40-s − 41-s + 45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2895 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2895 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.129036306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.129036306\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 193 | \( 1 - T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )^{2} \) |
| 31 | \( ( 1 - T )^{2} \) |
| 37 | \( ( 1 - T )^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 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
−8.883828300870245881352347576061, −8.314277000990984573003268104388, −7.77134362351763762218447252120, −6.98520762192062370703907005578, −6.07177073117937207327414395371, −4.85564438809103076935243447141, −4.41346764038031250355329043338, −2.79912450719429610279424421777, −2.36111872696878798492233569108, −1.14822065302048317076306458998,
1.14822065302048317076306458998, 2.36111872696878798492233569108, 2.79912450719429610279424421777, 4.41346764038031250355329043338, 4.85564438809103076935243447141, 6.07177073117937207327414395371, 6.98520762192062370703907005578, 7.77134362351763762218447252120, 8.314277000990984573003268104388, 8.883828300870245881352347576061
