L(s) = 1 | − 3-s + 3·5-s − 2·7-s − 2·9-s − 3·11-s + 5·13-s − 3·15-s − 4·17-s + 2·21-s + 4·25-s + 5·27-s + 29-s − 9·31-s + 3·33-s − 6·35-s − 8·37-s − 5·39-s − 2·41-s − 11·43-s − 6·45-s + 7·47-s − 3·49-s + 4·51-s − 9·53-s − 9·55-s + 4·59-s + 12·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.34·5-s − 0.755·7-s − 2/3·9-s − 0.904·11-s + 1.38·13-s − 0.774·15-s − 0.970·17-s + 0.436·21-s + 4/5·25-s + 0.962·27-s + 0.185·29-s − 1.61·31-s + 0.522·33-s − 1.01·35-s − 1.31·37-s − 0.800·39-s − 0.312·41-s − 1.67·43-s − 0.894·45-s + 1.02·47-s − 3/7·49-s + 0.560·51-s − 1.23·53-s − 1.21·55-s + 0.520·59-s + 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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
−8.868470335391324440586597620218, −8.317920725627485157632811027240, −6.88127849884191867875357055362, −6.39208865343620833912323561954, −5.59484198644236369333229457444, −5.18505606705021347413228227439, −3.71393366615299243699613634856, −2.73069587768107365355477934045, −1.68175753467464429238376356886, 0,
1.68175753467464429238376356886, 2.73069587768107365355477934045, 3.71393366615299243699613634856, 5.18505606705021347413228227439, 5.59484198644236369333229457444, 6.39208865343620833912323561954, 6.88127849884191867875357055362, 8.317920725627485157632811027240, 8.868470335391324440586597620218