| L(s) = 1 | − 3·3-s + 2·5-s − 7·7-s + 9·9-s + 12·11-s + 66·13-s − 6·15-s − 70·17-s − 92·19-s + 21·21-s − 16·23-s − 121·25-s − 27·27-s + 122·29-s − 64·31-s − 36·33-s − 14·35-s + 306·37-s − 198·39-s + 50·41-s + 20·43-s + 18·45-s + 176·47-s + 49·49-s + 210·51-s − 526·53-s + 24·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.178·5-s − 0.377·7-s + 1/3·9-s + 0.328·11-s + 1.40·13-s − 0.103·15-s − 0.998·17-s − 1.11·19-s + 0.218·21-s − 0.145·23-s − 0.967·25-s − 0.192·27-s + 0.781·29-s − 0.370·31-s − 0.189·33-s − 0.0676·35-s + 1.35·37-s − 0.812·39-s + 0.190·41-s + 0.0709·43-s + 0.0596·45-s + 0.546·47-s + 1/7·49-s + 0.576·51-s − 1.36·53-s + 0.0588·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 66 T + p^{3} T^{2} \) |
| 17 | \( 1 + 70 T + p^{3} T^{2} \) |
| 19 | \( 1 + 92 T + p^{3} T^{2} \) |
| 23 | \( 1 + 16 T + p^{3} T^{2} \) |
| 29 | \( 1 - 122 T + p^{3} T^{2} \) |
| 31 | \( 1 + 64 T + p^{3} T^{2} \) |
| 37 | \( 1 - 306 T + p^{3} T^{2} \) |
| 41 | \( 1 - 50 T + p^{3} T^{2} \) |
| 43 | \( 1 - 20 T + p^{3} T^{2} \) |
| 47 | \( 1 - 176 T + p^{3} T^{2} \) |
| 53 | \( 1 + 526 T + p^{3} T^{2} \) |
| 59 | \( 1 - 540 T + p^{3} T^{2} \) |
| 61 | \( 1 - 818 T + p^{3} T^{2} \) |
| 67 | \( 1 + 228 T + p^{3} T^{2} \) |
| 71 | \( 1 + 864 T + p^{3} T^{2} \) |
| 73 | \( 1 - 106 T + p^{3} T^{2} \) |
| 79 | \( 1 + 736 T + p^{3} T^{2} \) |
| 83 | \( 1 + 588 T + p^{3} T^{2} \) |
| 89 | \( 1 - 146 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1214 T + p^{3} T^{2} \) |
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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
−8.834575220262983325190901822303, −8.150596851337489740399150682310, −6.94884681493429737468508628367, −6.27287468035830831858854474243, −5.74989394142820937611258061535, −4.43728169326527915334347418401, −3.82656381447494471408577820096, −2.44634889041612884976384663111, −1.27023125744468206168894870452, 0,
1.27023125744468206168894870452, 2.44634889041612884976384663111, 3.82656381447494471408577820096, 4.43728169326527915334347418401, 5.74989394142820937611258061535, 6.27287468035830831858854474243, 6.94884681493429737468508628367, 8.150596851337489740399150682310, 8.834575220262983325190901822303
