| L(s) = 1 | + 3-s + 4·7-s + 9-s − 11-s − 13-s − 17-s − 2·19-s + 4·21-s − 4·23-s − 5·25-s + 27-s − 2·29-s − 33-s + 10·37-s − 39-s − 2·41-s + 6·43-s − 6·47-s + 9·49-s − 51-s + 2·53-s − 2·57-s + 10·59-s + 8·61-s + 4·63-s − 4·69-s − 4·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.242·17-s − 0.458·19-s + 0.872·21-s − 0.834·23-s − 25-s + 0.192·27-s − 0.371·29-s − 0.174·33-s + 1.64·37-s − 0.160·39-s − 0.312·41-s + 0.914·43-s − 0.875·47-s + 9/7·49-s − 0.140·51-s + 0.274·53-s − 0.264·57-s + 1.30·59-s + 1.02·61-s + 0.503·63-s − 0.481·69-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.98371441852241, −13.35309588341166, −12.96293525791466, −12.45363696740018, −11.76788134094072, −11.32068827801449, −11.17187925521115, −10.28069403642891, −10.00153312195302, −9.454955604258528, −8.764462700043994, −8.348437016574942, −7.984658978214248, −7.512021478065449, −7.073781492786277, −6.262545604534386, −5.719505667250006, −5.216456127725980, −4.505319338234103, −4.202120269259890, −3.638364351613147, −2.676984332918105, −2.255720569769199, −1.739076793802057, −1.023315417503405, 0,
1.023315417503405, 1.739076793802057, 2.255720569769199, 2.676984332918105, 3.638364351613147, 4.202120269259890, 4.505319338234103, 5.216456127725980, 5.719505667250006, 6.262545604534386, 7.073781492786277, 7.512021478065449, 7.984658978214248, 8.348437016574942, 8.764462700043994, 9.454955604258528, 10.00153312195302, 10.28069403642891, 11.17187925521115, 11.32068827801449, 11.76788134094072, 12.45363696740018, 12.96293525791466, 13.35309588341166, 13.98371441852241
