L(s) = 1 | + 2·5-s − 2·7-s + 2·11-s + 13-s − 6·17-s + 2·19-s − 25-s + 2·29-s + 2·31-s − 4·35-s − 10·37-s − 2·41-s − 8·43-s + 2·47-s − 3·49-s − 2·53-s + 4·55-s + 10·59-s − 6·61-s + 2·65-s − 14·67-s + 14·71-s − 6·73-s − 4·77-s − 4·79-s + 6·83-s − 12·85-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.755·7-s + 0.603·11-s + 0.277·13-s − 1.45·17-s + 0.458·19-s − 1/5·25-s + 0.371·29-s + 0.359·31-s − 0.676·35-s − 1.64·37-s − 0.312·41-s − 1.21·43-s + 0.291·47-s − 3/7·49-s − 0.274·53-s + 0.539·55-s + 1.30·59-s − 0.768·61-s + 0.248·65-s − 1.71·67-s + 1.66·71-s − 0.702·73-s − 0.455·77-s − 0.450·79-s + 0.658·83-s − 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.38531180908039202565647074162, −6.55596306495327967863391718811, −6.41020350393506582534491295126, −5.49905970191346321747782732985, −4.77121810409486383912343844767, −3.86646896295917195322554897219, −3.12346547114610052261363571950, −2.19558529713056305604103152017, −1.41046291764702975902444090051, 0,
1.41046291764702975902444090051, 2.19558529713056305604103152017, 3.12346547114610052261363571950, 3.86646896295917195322554897219, 4.77121810409486383912343844767, 5.49905970191346321747782732985, 6.41020350393506582534491295126, 6.55596306495327967863391718811, 7.38531180908039202565647074162