| L(s) = 1 | − 5-s + 7-s − 3·9-s − 6·11-s + 6·13-s + 7·17-s − 2·19-s + 23-s + 25-s − 5·29-s − 31-s − 35-s − 5·37-s − 7·41-s − 8·43-s + 3·45-s − 8·47-s − 6·49-s + 3·53-s + 6·55-s − 13·59-s − 8·61-s − 3·63-s − 6·65-s + 9·67-s − 7·71-s − 2·73-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 9-s − 1.80·11-s + 1.66·13-s + 1.69·17-s − 0.458·19-s + 0.208·23-s + 1/5·25-s − 0.928·29-s − 0.179·31-s − 0.169·35-s − 0.821·37-s − 1.09·41-s − 1.21·43-s + 0.447·45-s − 1.16·47-s − 6/7·49-s + 0.412·53-s + 0.809·55-s − 1.69·59-s − 1.02·61-s − 0.377·63-s − 0.744·65-s + 1.09·67-s − 0.830·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−8.490339300623037157281889260704, −8.213660532125821765464224346923, −7.54352683872682538729110646066, −6.34632115168232953381843862578, −5.50548101080189124840642085625, −4.99241813943990397520918318295, −3.55437512572495651147208683365, −3.05356706831540426127652662338, −1.61688740435185646490190999504, 0,
1.61688740435185646490190999504, 3.05356706831540426127652662338, 3.55437512572495651147208683365, 4.99241813943990397520918318295, 5.50548101080189124840642085625, 6.34632115168232953381843862578, 7.54352683872682538729110646066, 8.213660532125821765464224346923, 8.490339300623037157281889260704
