| L(s) = 1 | − 96.7·5-s + 49·7-s + 281.·11-s − 269.·13-s − 1.71e3·17-s + 1.17e3·19-s + 785.·23-s + 6.23e3·25-s − 6.14e3·29-s + 7.00e3·31-s − 4.73e3·35-s − 1.14e4·37-s + 1.32e4·41-s + 1.98e4·43-s + 6.88e3·47-s + 2.40e3·49-s − 8.65e3·53-s − 2.72e4·55-s − 4.78e4·59-s + 5.27e4·61-s + 2.60e4·65-s + 2.40e4·67-s + 1.25e4·71-s + 4.07e3·73-s + 1.37e4·77-s + 1.19e4·79-s + 8.19e4·83-s + ⋯ |
| L(s) = 1 | − 1.73·5-s + 0.377·7-s + 0.701·11-s − 0.442·13-s − 1.44·17-s + 0.745·19-s + 0.309·23-s + 1.99·25-s − 1.35·29-s + 1.30·31-s − 0.653·35-s − 1.38·37-s + 1.23·41-s + 1.63·43-s + 0.454·47-s + 0.142·49-s − 0.423·53-s − 1.21·55-s − 1.78·59-s + 1.81·61-s + 0.764·65-s + 0.654·67-s + 0.295·71-s + 0.0895·73-s + 0.265·77-s + 0.214·79-s + 1.30·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 49T \) |
| good | 5 | \( 1 + 96.7T + 3.12e3T^{2} \) |
| 11 | \( 1 - 281.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 269.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.71e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.17e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 785.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.14e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.00e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.14e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.32e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.98e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.88e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 8.65e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.78e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.27e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.40e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.25e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.07e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.19e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.69e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.64e4T + 8.58e9T^{2} \) |
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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
−8.788583736632121018759224213150, −7.86306698847164462236556414466, −7.30992995182567865005322322296, −6.48140196088266776150910194411, −5.11106346973371617143128620697, −4.28189411681381673905601221170, −3.65515025034331237826678027192, −2.44900018590490422955780581196, −1.00244756956887179034387628849, 0,
1.00244756956887179034387628849, 2.44900018590490422955780581196, 3.65515025034331237826678027192, 4.28189411681381673905601221170, 5.11106346973371617143128620697, 6.48140196088266776150910194411, 7.30992995182567865005322322296, 7.86306698847164462236556414466, 8.788583736632121018759224213150
