| L(s) = 1 | − 3-s + 5·7-s + 9-s + 11-s + 4·13-s − 5·17-s + 7·19-s − 5·21-s + 9·23-s − 27-s + 2·29-s + 4·31-s − 33-s − 7·37-s − 4·39-s − 7·41-s − 9·47-s + 18·49-s + 5·51-s + 2·53-s − 7·57-s − 7·59-s + 2·61-s + 5·63-s + 2·67-s − 9·69-s + 5·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.88·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 1.21·17-s + 1.60·19-s − 1.09·21-s + 1.87·23-s − 0.192·27-s + 0.371·29-s + 0.718·31-s − 0.174·33-s − 1.15·37-s − 0.640·39-s − 1.09·41-s − 1.31·47-s + 18/7·49-s + 0.700·51-s + 0.274·53-s − 0.927·57-s − 0.911·59-s + 0.256·61-s + 0.629·63-s + 0.244·67-s − 1.08·69-s + 0.593·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.276148410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.276148410\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.627439385745066736570736973323, −7.944958395163614005497572811874, −7.07656863122636208710945008694, −6.47938585590135417751307146358, −5.29078075784349630875638328850, −5.01478235399514540995977256515, −4.16691336013417842305471402128, −3.07316015217269848452851586644, −1.69089768405538642612634766548, −1.05879644449581225375370482134,
1.05879644449581225375370482134, 1.69089768405538642612634766548, 3.07316015217269848452851586644, 4.16691336013417842305471402128, 5.01478235399514540995977256515, 5.29078075784349630875638328850, 6.47938585590135417751307146358, 7.07656863122636208710945008694, 7.944958395163614005497572811874, 8.627439385745066736570736973323
