| L(s) = 1 | + 5-s − 3·9-s + 17-s + 4·19-s − 8·23-s + 25-s + 2·29-s + 8·31-s − 2·37-s − 2·41-s − 4·43-s − 3·45-s − 7·49-s + 6·53-s + 4·59-s − 6·61-s − 4·67-s + 8·71-s − 2·73-s + 9·81-s − 4·83-s + 85-s + 6·89-s + 4·95-s − 18·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 9-s + 0.242·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.328·37-s − 0.312·41-s − 0.609·43-s − 0.447·45-s − 49-s + 0.824·53-s + 0.520·59-s − 0.768·61-s − 0.488·67-s + 0.949·71-s − 0.234·73-s + 81-s − 0.439·83-s + 0.108·85-s + 0.635·89-s + 0.410·95-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.76449461470202, −13.63602835488539, −12.93090116196849, −12.16584531681658, −12.02271615814958, −11.48074352193615, −11.03286423757002, −10.20608252902466, −10.05701066421385, −9.554531110141826, −8.858267160519780, −8.441282378779458, −7.982753078307330, −7.495578687214611, −6.732676813019937, −6.249951034562105, −5.855298204363358, −5.255012726537873, −4.818484880368976, −4.076017004823252, −3.411043564373666, −2.909043280832863, −2.309686815922118, −1.638601721052245, −0.8519591088031592, 0,
0.8519591088031592, 1.638601721052245, 2.309686815922118, 2.909043280832863, 3.411043564373666, 4.076017004823252, 4.818484880368976, 5.255012726537873, 5.855298204363358, 6.249951034562105, 6.732676813019937, 7.495578687214611, 7.982753078307330, 8.441282378779458, 8.858267160519780, 9.554531110141826, 10.05701066421385, 10.20608252902466, 11.03286423757002, 11.48074352193615, 12.02271615814958, 12.16584531681658, 12.93090116196849, 13.63602835488539, 13.76449461470202
