| L(s) = 1 | + 5-s − 7-s − 4·13-s − 17-s + 4·23-s − 4·25-s + 2·31-s − 35-s + 6·37-s + 2·41-s + 3·43-s + 7·47-s + 49-s − 12·53-s − 5·59-s − 12·61-s − 4·65-s − 5·67-s − 6·71-s + 2·73-s + 8·79-s + 15·83-s − 85-s − 9·89-s + 4·91-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 1.10·13-s − 0.242·17-s + 0.834·23-s − 4/5·25-s + 0.359·31-s − 0.169·35-s + 0.986·37-s + 0.312·41-s + 0.457·43-s + 1.02·47-s + 1/7·49-s − 1.64·53-s − 0.650·59-s − 1.53·61-s − 0.496·65-s − 0.610·67-s − 0.712·71-s + 0.234·73-s + 0.900·79-s + 1.64·83-s − 0.108·85-s − 0.953·89-s + 0.419·91-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.63723136819271, −13.46493446947985, −12.73615029406545, −12.42127118455168, −11.94870639148601, −11.39572964693882, −10.71452976336388, −10.53396788396371, −9.735670570951621, −9.365249281899139, −9.205446067990797, −8.362840865619339, −7.745587313090844, −7.443699507210246, −6.808421656993664, −6.227331813868968, −5.863726217821075, −5.211022224088592, −4.594096722364813, −4.263045667655279, −3.349245056826911, −2.857816583357626, −2.305162025973633, −1.661682790043307, −0.8130550635033637, 0,
0.8130550635033637, 1.661682790043307, 2.305162025973633, 2.857816583357626, 3.349245056826911, 4.263045667655279, 4.594096722364813, 5.211022224088592, 5.863726217821075, 6.227331813868968, 6.808421656993664, 7.443699507210246, 7.745587313090844, 8.362840865619339, 9.205446067990797, 9.365249281899139, 9.735670570951621, 10.53396788396371, 10.71452976336388, 11.39572964693882, 11.94870639148601, 12.42127118455168, 12.73615029406545, 13.46493446947985, 13.63723136819271
