L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 4·7-s + 8-s + 9-s + 11-s − 12-s − 13-s − 4·14-s + 16-s − 17-s + 18-s + 2·19-s + 4·21-s + 22-s + 4·23-s − 24-s − 5·25-s − 26-s − 27-s − 4·28-s − 2·29-s + 32-s − 33-s − 34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 0.277·13-s − 1.06·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.458·19-s + 0.872·21-s + 0.213·22-s + 0.834·23-s − 0.204·24-s − 25-s − 0.196·26-s − 0.192·27-s − 0.755·28-s − 0.371·29-s + 0.176·32-s − 0.174·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 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
−16.42483644435286, −15.80836152203658, −15.24329934673066, −14.87472244015863, −13.96530964104911, −13.44440007244202, −13.04515749839400, −12.50710401568393, −11.93807044381214, −11.46022521829662, −10.78477249638961, −10.16615515671976, −9.496703106984112, −9.243832224375774, −8.134513612697486, −7.397210815631041, −6.821069128920514, −6.322708551914773, −5.769274058779275, −5.136870029500918, −4.308060255601427, −3.689272798012281, −3.018616371400493, −2.256992340238214, −1.087141702718866, 0,
1.087141702718866, 2.256992340238214, 3.018616371400493, 3.689272798012281, 4.308060255601427, 5.136870029500918, 5.769274058779275, 6.322708551914773, 6.821069128920514, 7.397210815631041, 8.134513612697486, 9.243832224375774, 9.496703106984112, 10.16615515671976, 10.78477249638961, 11.46022521829662, 11.93807044381214, 12.50710401568393, 13.04515749839400, 13.44440007244202, 13.96530964104911, 14.87472244015863, 15.24329934673066, 15.80836152203658, 16.42483644435286