| L(s) = 1 | − 2·3-s − 3·7-s + 9-s + 3·11-s + 4·13-s − 5·17-s + 19-s + 6·21-s + 4·27-s + 2·29-s − 8·31-s − 6·33-s + 10·37-s − 8·39-s + 6·41-s − 7·43-s − 9·47-s + 2·49-s + 10·51-s + 8·53-s − 2·57-s − 14·59-s − 5·61-s − 3·63-s + 6·71-s + 15·73-s − 9·77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.13·7-s + 1/3·9-s + 0.904·11-s + 1.10·13-s − 1.21·17-s + 0.229·19-s + 1.30·21-s + 0.769·27-s + 0.371·29-s − 1.43·31-s − 1.04·33-s + 1.64·37-s − 1.28·39-s + 0.937·41-s − 1.06·43-s − 1.31·47-s + 2/7·49-s + 1.40·51-s + 1.09·53-s − 0.264·57-s − 1.82·59-s − 0.640·61-s − 0.377·63-s + 0.712·71-s + 1.75·73-s − 1.02·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8334784344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8334784344\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−7.80595480702685947773424134871, −6.68553027648366020145094881467, −6.53379416833881480288868750043, −5.98018473233793983696353323958, −5.20785389701441733439865635683, −4.30567049794628832835810234888, −3.66114900132282461221025518109, −2.79145226836229010963645843021, −1.53781451857294495399244090671, −0.50030718477405811673021345760,
0.50030718477405811673021345760, 1.53781451857294495399244090671, 2.79145226836229010963645843021, 3.66114900132282461221025518109, 4.30567049794628832835810234888, 5.20785389701441733439865635683, 5.98018473233793983696353323958, 6.53379416833881480288868750043, 6.68553027648366020145094881467, 7.80595480702685947773424134871
