| L(s) = 1 | + 2-s + 4-s − 5-s + 3·7-s + 8-s − 3·9-s − 10-s − 11-s + 13-s + 3·14-s + 16-s − 4·17-s − 3·18-s − 7·19-s − 20-s − 22-s + 4·23-s + 25-s + 26-s + 3·28-s + 3·29-s − 9·31-s + 32-s − 4·34-s − 3·35-s − 3·36-s − 4·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.13·7-s + 0.353·8-s − 9-s − 0.316·10-s − 0.301·11-s + 0.277·13-s + 0.801·14-s + 1/4·16-s − 0.970·17-s − 0.707·18-s − 1.60·19-s − 0.223·20-s − 0.213·22-s + 0.834·23-s + 1/5·25-s + 0.196·26-s + 0.566·28-s + 0.557·29-s − 1.61·31-s + 0.176·32-s − 0.685·34-s − 0.507·35-s − 1/2·36-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−8.008685432435982464738465787525, −7.11164799797501987751183011094, −6.44519273873947521282882141562, −5.58436979995921188363024271037, −4.90464452175472170244343735019, −4.31140617473637365416974181277, −3.41377703753420460862047856184, −2.48002386039788251025702622555, −1.65726053451929272510225814310, 0,
1.65726053451929272510225814310, 2.48002386039788251025702622555, 3.41377703753420460862047856184, 4.31140617473637365416974181277, 4.90464452175472170244343735019, 5.58436979995921188363024271037, 6.44519273873947521282882141562, 7.11164799797501987751183011094, 8.008685432435982464738465787525
