| L(s) = 1 | − 2·7-s − 3·9-s + 4·11-s + 4·13-s − 6·17-s − 19-s − 2·23-s − 6·29-s + 8·31-s − 4·37-s + 6·41-s − 6·43-s + 6·47-s − 3·49-s − 8·53-s + 12·59-s + 6·61-s + 6·63-s + 10·73-s − 8·77-s + 8·79-s + 9·81-s + 14·83-s + 14·89-s − 8·91-s − 16·97-s − 12·99-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 9-s + 1.20·11-s + 1.10·13-s − 1.45·17-s − 0.229·19-s − 0.417·23-s − 1.11·29-s + 1.43·31-s − 0.657·37-s + 0.937·41-s − 0.914·43-s + 0.875·47-s − 3/7·49-s − 1.09·53-s + 1.56·59-s + 0.768·61-s + 0.755·63-s + 1.17·73-s − 0.911·77-s + 0.900·79-s + 81-s + 1.53·83-s + 1.48·89-s − 0.838·91-s − 1.62·97-s − 1.20·99-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
| \(L(1)\) |
\(\approx\) |
\(1.472989491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.472989491\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−8.139992078522444116281483990501, −6.87938551537092652524838647013, −6.48004545665269494951382916541, −6.01534214212804574435785232311, −5.13073410322056411249741057685, −4.06904521426402208328109551785, −3.66750238038915502862341443592, −2.72632954691590884043355037420, −1.82893705040273253134830073786, −0.59630071697762105413554919012,
0.59630071697762105413554919012, 1.82893705040273253134830073786, 2.72632954691590884043355037420, 3.66750238038915502862341443592, 4.06904521426402208328109551785, 5.13073410322056411249741057685, 6.01534214212804574435785232311, 6.48004545665269494951382916541, 6.87938551537092652524838647013, 8.139992078522444116281483990501
