L(s) = 1 | − 3-s + 5-s + 9-s − 11-s + 13-s − 15-s − 4·17-s − 6·19-s − 23-s + 25-s − 27-s + 5·29-s + 33-s − 39-s − 3·41-s + 43-s + 45-s − 7·47-s + 4·51-s − 3·53-s − 55-s + 6·57-s − 8·59-s + 6·61-s + 65-s + 14·67-s + 69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s + 0.277·13-s − 0.258·15-s − 0.970·17-s − 1.37·19-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.928·29-s + 0.174·33-s − 0.160·39-s − 0.468·41-s + 0.152·43-s + 0.149·45-s − 1.02·47-s + 0.560·51-s − 0.412·53-s − 0.134·55-s + 0.794·57-s − 1.04·59-s + 0.768·61-s + 0.124·65-s + 1.71·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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
−15.41267650276420, −14.81529320033364, −14.23732898350052, −13.60530015488128, −13.21926576048631, −12.62402474950037, −12.26627674547459, −11.49577236413436, −10.94828502026113, −10.66192512814796, −10.00289100494079, −9.489742540476247, −8.802095557684405, −8.288824940596881, −7.756123075628178, −6.770640220733496, −6.497585364645720, −6.072681951434813, −5.122343767883495, −4.840037384005664, −4.068155787857129, −3.395956576289063, −2.362485671958040, −1.980997385034965, −0.9301383370382955, 0,
0.9301383370382955, 1.980997385034965, 2.362485671958040, 3.395956576289063, 4.068155787857129, 4.840037384005664, 5.122343767883495, 6.072681951434813, 6.497585364645720, 6.770640220733496, 7.756123075628178, 8.288824940596881, 8.802095557684405, 9.489742540476247, 10.00289100494079, 10.66192512814796, 10.94828502026113, 11.49577236413436, 12.26627674547459, 12.62402474950037, 13.21926576048631, 13.60530015488128, 14.23732898350052, 14.81529320033364, 15.41267650276420