| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 6·11-s − 2·13-s − 14-s + 16-s − 4·19-s − 6·22-s − 9·23-s + 2·26-s + 28-s + 3·29-s − 4·31-s − 32-s − 8·37-s + 4·38-s − 3·41-s − 8·43-s + 6·44-s + 9·46-s + 3·47-s − 6·49-s − 2·52-s − 6·53-s − 56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 1.80·11-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.917·19-s − 1.27·22-s − 1.87·23-s + 0.392·26-s + 0.188·28-s + 0.557·29-s − 0.718·31-s − 0.176·32-s − 1.31·37-s + 0.648·38-s − 0.468·41-s − 1.21·43-s + 0.904·44-s + 1.32·46-s + 0.437·47-s − 6/7·49-s − 0.277·52-s − 0.824·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.248433259321795466549359074409, −7.40092613145234491403247897085, −6.59217107502750007616724668699, −6.19247399766751704124881422073, −5.07336105369884447208985677977, −4.15590134142101499861041228246, −3.44863339065223970410846496972, −2.09429920926468511704223980474, −1.49500107253082459697348398877, 0,
1.49500107253082459697348398877, 2.09429920926468511704223980474, 3.44863339065223970410846496972, 4.15590134142101499861041228246, 5.07336105369884447208985677977, 6.19247399766751704124881422073, 6.59217107502750007616724668699, 7.40092613145234491403247897085, 8.248433259321795466549359074409
