| L(s) = 1 | − 3-s − 7-s − 2·9-s − 3·11-s + 4·13-s + 6·17-s − 2·19-s + 21-s + 6·23-s − 5·25-s + 5·27-s + 6·29-s − 4·31-s + 3·33-s − 37-s − 4·39-s − 9·41-s − 8·43-s + 3·47-s − 6·49-s − 6·51-s + 3·53-s + 2·57-s − 12·59-s − 8·61-s + 2·63-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s − 0.904·11-s + 1.10·13-s + 1.45·17-s − 0.458·19-s + 0.218·21-s + 1.25·23-s − 25-s + 0.962·27-s + 1.11·29-s − 0.718·31-s + 0.522·33-s − 0.164·37-s − 0.640·39-s − 1.40·41-s − 1.21·43-s + 0.437·47-s − 6/7·49-s − 0.840·51-s + 0.412·53-s + 0.264·57-s − 1.56·59-s − 1.02·61-s + 0.251·63-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.487594403486838813155108109695, −7.953080976467459568741615470163, −6.92702526110685059626904983928, −6.13026790010533892031415629049, −5.52397436821147602871559735263, −4.81124860316907390767416903100, −3.49656973901712201010716232454, −2.90414738025441870892963779928, −1.39370194281586028856128276808, 0,
1.39370194281586028856128276808, 2.90414738025441870892963779928, 3.49656973901712201010716232454, 4.81124860316907390767416903100, 5.52397436821147602871559735263, 6.13026790010533892031415629049, 6.92702526110685059626904983928, 7.953080976467459568741615470163, 8.487594403486838813155108109695
