L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s − 3·7-s + 9-s + 2·12-s + 13-s − 6·14-s − 4·16-s + 2·17-s + 2·18-s + 5·19-s − 3·21-s − 6·23-s + 2·26-s + 27-s − 6·28-s − 10·29-s − 3·31-s − 8·32-s + 4·34-s + 2·36-s − 2·37-s + 10·38-s + 39-s + 8·41-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s − 1.13·7-s + 1/3·9-s + 0.577·12-s + 0.277·13-s − 1.60·14-s − 16-s + 0.485·17-s + 0.471·18-s + 1.14·19-s − 0.654·21-s − 1.25·23-s + 0.392·26-s + 0.192·27-s − 1.13·28-s − 1.85·29-s − 0.538·31-s − 1.41·32-s + 0.685·34-s + 1/3·36-s − 0.328·37-s + 1.62·38-s + 0.160·39-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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
−7.34923234321660315078497154764, −6.47550891297445546652062110512, −5.83602188561825449212965209319, −5.44264591099408884558544576394, −4.38637866918120839883207984982, −3.75734065175266443138175607180, −3.28631221908682607371983175186, −2.62960787106095443058504699500, −1.60835528081174442082226795369, 0,
1.60835528081174442082226795369, 2.62960787106095443058504699500, 3.28631221908682607371983175186, 3.75734065175266443138175607180, 4.38637866918120839883207984982, 5.44264591099408884558544576394, 5.83602188561825449212965209319, 6.47550891297445546652062110512, 7.34923234321660315078497154764