| L(s) = 1 | + 5-s + 7-s + 11-s − 5·13-s + 6·17-s − 19-s − 4·23-s − 4·25-s − 29-s − 10·31-s + 35-s + 37-s − 8·43-s − 47-s + 49-s − 8·53-s + 55-s + 3·59-s − 6·61-s − 5·65-s + 13·67-s − 3·73-s + 77-s − 8·79-s − 2·83-s + 6·85-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 0.301·11-s − 1.38·13-s + 1.45·17-s − 0.229·19-s − 0.834·23-s − 4/5·25-s − 0.185·29-s − 1.79·31-s + 0.169·35-s + 0.164·37-s − 1.21·43-s − 0.145·47-s + 1/7·49-s − 1.09·53-s + 0.134·55-s + 0.390·59-s − 0.768·61-s − 0.620·65-s + 1.58·67-s − 0.351·73-s + 0.113·77-s − 0.900·79-s − 0.219·83-s + 0.650·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.71402067755677122990686690521, −7.24345378117054855121122671969, −6.29847709010423726295652730463, −5.54158002770924338290083506400, −5.04590168525471106637079743564, −4.07307015615996649673895259934, −3.27731109362979123195070337450, −2.21198238526017587895740189491, −1.51950365234421232982822568109, 0,
1.51950365234421232982822568109, 2.21198238526017587895740189491, 3.27731109362979123195070337450, 4.07307015615996649673895259934, 5.04590168525471106637079743564, 5.54158002770924338290083506400, 6.29847709010423726295652730463, 7.24345378117054855121122671969, 7.71402067755677122990686690521
