L(s) = 1 | + 3-s − 2·4-s + 5-s − 2·7-s + 9-s + 3·11-s − 2·12-s − 5·13-s + 15-s + 4·16-s + 17-s + 3·19-s − 2·20-s − 2·21-s − 23-s − 4·25-s + 27-s + 4·28-s + 2·29-s − 4·31-s + 3·33-s − 2·35-s − 2·36-s − 8·37-s − 5·39-s − 11·41-s + 3·43-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.904·11-s − 0.577·12-s − 1.38·13-s + 0.258·15-s + 16-s + 0.242·17-s + 0.688·19-s − 0.447·20-s − 0.436·21-s − 0.208·23-s − 4/5·25-s + 0.192·27-s + 0.755·28-s + 0.371·29-s − 0.718·31-s + 0.522·33-s − 0.338·35-s − 1/3·36-s − 1.31·37-s − 0.800·39-s − 1.71·41-s + 0.457·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 + T \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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
−8.182839741152360142350568454038, −7.35239004178596219728736686128, −6.74897525159459081013893232923, −5.68715351380410781538432457072, −5.11839517352071854249384785032, −4.09867772040793191300725564433, −3.52200200188685535063006636750, −2.56848790550944790468601802880, −1.42301478666430522235507950230, 0,
1.42301478666430522235507950230, 2.56848790550944790468601802880, 3.52200200188685535063006636750, 4.09867772040793191300725564433, 5.11839517352071854249384785032, 5.68715351380410781538432457072, 6.74897525159459081013893232923, 7.35239004178596219728736686128, 8.182839741152360142350568454038