L(s) = 1 | − 3-s + 7-s + 9-s + 2·11-s − 4·13-s + 19-s − 21-s − 6·23-s − 5·25-s − 27-s + 2·29-s + 4·31-s − 2·33-s + 2·37-s + 4·39-s + 2·41-s − 8·43-s − 8·47-s + 49-s − 6·53-s − 57-s − 4·59-s + 14·61-s + 63-s − 2·67-s + 6·69-s − 6·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s + 0.229·19-s − 0.218·21-s − 1.25·23-s − 25-s − 0.192·27-s + 0.371·29-s + 0.718·31-s − 0.348·33-s + 0.328·37-s + 0.640·39-s + 0.312·41-s − 1.21·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s − 0.132·57-s − 0.520·59-s + 1.79·61-s + 0.125·63-s − 0.244·67-s + 0.722·69-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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
−8.126728822965856872245344268629, −7.61449736836411210829257604926, −6.68471916605767197977812703357, −6.08600051705823631580745227099, −5.17388322184790843889986275291, −4.52107706135484841599095676545, −3.67269445153567228289988183780, −2.43859040498123804703949075559, −1.44599544060078995492712723228, 0,
1.44599544060078995492712723228, 2.43859040498123804703949075559, 3.67269445153567228289988183780, 4.52107706135484841599095676545, 5.17388322184790843889986275291, 6.08600051705823631580745227099, 6.68471916605767197977812703357, 7.61449736836411210829257604926, 8.126728822965856872245344268629