L(s) = 1 | + 2-s − 0.198·3-s + 4-s + 0.890·5-s − 0.198·6-s − 7-s + 8-s − 2.96·9-s + 0.890·10-s + 0.664·11-s − 0.198·12-s − 14-s − 0.176·15-s + 16-s − 2.44·17-s − 2.96·18-s − 8.63·19-s + 0.890·20-s + 0.198·21-s + 0.664·22-s − 7.60·23-s − 0.198·24-s − 4.20·25-s + 1.18·27-s − 28-s + 3.60·29-s − 0.176·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.114·3-s + 0.5·4-s + 0.398·5-s − 0.0808·6-s − 0.377·7-s + 0.353·8-s − 0.986·9-s + 0.281·10-s + 0.200·11-s − 0.0571·12-s − 0.267·14-s − 0.0455·15-s + 0.250·16-s − 0.593·17-s − 0.697·18-s − 1.98·19-s + 0.199·20-s + 0.0432·21-s + 0.141·22-s − 1.58·23-s − 0.0404·24-s − 0.841·25-s + 0.227·27-s − 0.188·28-s + 0.669·29-s − 0.0321·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 0.198T + 3T^{2} \) |
| 5 | \( 1 - 0.890T + 5T^{2} \) |
| 11 | \( 1 - 0.664T + 11T^{2} \) |
| 17 | \( 1 + 2.44T + 17T^{2} \) |
| 19 | \( 1 + 8.63T + 19T^{2} \) |
| 23 | \( 1 + 7.60T + 23T^{2} \) |
| 29 | \( 1 - 3.60T + 29T^{2} \) |
| 31 | \( 1 + 1.50T + 31T^{2} \) |
| 37 | \( 1 + 5.60T + 37T^{2} \) |
| 41 | \( 1 - 7.83T + 41T^{2} \) |
| 43 | \( 1 - 7.46T + 43T^{2} \) |
| 47 | \( 1 + 1.20T + 47T^{2} \) |
| 53 | \( 1 + 4.89T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 - 7.70T + 61T^{2} \) |
| 67 | \( 1 - 6.07T + 67T^{2} \) |
| 71 | \( 1 + 6.27T + 71T^{2} \) |
| 73 | \( 1 - 3.67T + 73T^{2} \) |
| 79 | \( 1 - 4.37T + 79T^{2} \) |
| 83 | \( 1 + 2.92T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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
−8.553002730704364656247266730966, −7.85364217211133381955964297298, −6.68191337384785991318303201294, −6.14880098932720461896367868909, −5.63310137788663928691385174312, −4.47472768324780206959237674319, −3.83907833483421840763890757127, −2.65406921201735599756584591509, −1.95233551606990059219029172941, 0,
1.95233551606990059219029172941, 2.65406921201735599756584591509, 3.83907833483421840763890757127, 4.47472768324780206959237674319, 5.63310137788663928691385174312, 6.14880098932720461896367868909, 6.68191337384785991318303201294, 7.85364217211133381955964297298, 8.553002730704364656247266730966