| L(s) = 1 | + 4·5-s + 7·7-s + 62·11-s − 62·13-s − 84·17-s − 100·19-s − 42·23-s − 109·25-s + 10·29-s + 48·31-s + 28·35-s − 246·37-s + 248·41-s − 68·43-s + 324·47-s + 49·49-s − 258·53-s + 248·55-s + 120·59-s + 622·61-s − 248·65-s − 904·67-s − 678·71-s − 642·73-s + 434·77-s − 740·79-s + 468·83-s + ⋯ |
| L(s) = 1 | + 0.357·5-s + 0.377·7-s + 1.69·11-s − 1.32·13-s − 1.19·17-s − 1.20·19-s − 0.380·23-s − 0.871·25-s + 0.0640·29-s + 0.278·31-s + 0.135·35-s − 1.09·37-s + 0.944·41-s − 0.241·43-s + 1.00·47-s + 1/7·49-s − 0.668·53-s + 0.608·55-s + 0.264·59-s + 1.30·61-s − 0.473·65-s − 1.64·67-s − 1.13·71-s − 1.02·73-s + 0.642·77-s − 1.05·79-s + 0.618·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| good | 5 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 62 T + p^{3} T^{2} \) |
| 13 | \( 1 + 62 T + p^{3} T^{2} \) |
| 17 | \( 1 + 84 T + p^{3} T^{2} \) |
| 19 | \( 1 + 100 T + p^{3} T^{2} \) |
| 23 | \( 1 + 42 T + p^{3} T^{2} \) |
| 29 | \( 1 - 10 T + p^{3} T^{2} \) |
| 31 | \( 1 - 48 T + p^{3} T^{2} \) |
| 37 | \( 1 + 246 T + p^{3} T^{2} \) |
| 41 | \( 1 - 248 T + p^{3} T^{2} \) |
| 43 | \( 1 + 68 T + p^{3} T^{2} \) |
| 47 | \( 1 - 324 T + p^{3} T^{2} \) |
| 53 | \( 1 + 258 T + p^{3} T^{2} \) |
| 59 | \( 1 - 120 T + p^{3} T^{2} \) |
| 61 | \( 1 - 622 T + p^{3} T^{2} \) |
| 67 | \( 1 + 904 T + p^{3} T^{2} \) |
| 71 | \( 1 + 678 T + p^{3} T^{2} \) |
| 73 | \( 1 + 642 T + p^{3} T^{2} \) |
| 79 | \( 1 + 740 T + p^{3} T^{2} \) |
| 83 | \( 1 - 468 T + p^{3} T^{2} \) |
| 89 | \( 1 + 200 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1266 T + p^{3} T^{2} \) |
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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
−9.137743186179321710174272637302, −8.529415328332738701661956609111, −7.35958710126179500384683366875, −6.62940510998993838655441685977, −5.83080654846464504650839758236, −4.58765250890775039817825962885, −4.01127957021923969680805760266, −2.45850785702097720254556986714, −1.60259480098992732085903521435, 0,
1.60259480098992732085903521435, 2.45850785702097720254556986714, 4.01127957021923969680805760266, 4.58765250890775039817825962885, 5.83080654846464504650839758236, 6.62940510998993838655441685977, 7.35958710126179500384683366875, 8.529415328332738701661956609111, 9.137743186179321710174272637302
