| L(s) = 1 | + 3-s − 2·5-s + 9-s − 13-s − 2·15-s + 2·17-s − 4·19-s − 25-s + 27-s − 6·29-s + 2·37-s − 39-s + 6·41-s − 12·43-s − 2·45-s + 4·47-s − 7·49-s + 2·51-s − 6·53-s − 4·57-s − 8·59-s + 2·61-s + 2·65-s + 4·67-s + 12·71-s − 14·73-s − 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.277·13-s − 0.516·15-s + 0.485·17-s − 0.917·19-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.328·37-s − 0.160·39-s + 0.937·41-s − 1.82·43-s − 0.298·45-s + 0.583·47-s − 49-s + 0.280·51-s − 0.824·53-s − 0.529·57-s − 1.04·59-s + 0.256·61-s + 0.248·65-s + 0.488·67-s + 1.42·71-s − 1.63·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.330665390774691851001029176991, −7.919591009438183231618714407175, −7.18473757011111228492705194977, −6.34521314940595313311640841587, −5.30609149044046423898665363243, −4.33170052924158095404123814794, −3.69707925837485827655820809477, −2.77951541453721908240957674341, −1.63875743848026916087653140890, 0,
1.63875743848026916087653140890, 2.77951541453721908240957674341, 3.69707925837485827655820809477, 4.33170052924158095404123814794, 5.30609149044046423898665363243, 6.34521314940595313311640841587, 7.18473757011111228492705194977, 7.919591009438183231618714407175, 8.330665390774691851001029176991
