| L(s) = 1 | − 7-s − 3·9-s + 4·11-s − 13-s − 3·17-s − 8·19-s − 23-s − 8·29-s − 4·31-s + 5·37-s + 6·41-s − 2·43-s − 47-s + 49-s + 7·53-s + 3·63-s − 8·67-s + 12·71-s − 4·73-s − 4·77-s − 5·79-s + 9·81-s − 17·83-s − 3·89-s + 91-s − 13·97-s − 12·99-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s + 1.20·11-s − 0.277·13-s − 0.727·17-s − 1.83·19-s − 0.208·23-s − 1.48·29-s − 0.718·31-s + 0.821·37-s + 0.937·41-s − 0.304·43-s − 0.145·47-s + 1/7·49-s + 0.961·53-s + 0.377·63-s − 0.977·67-s + 1.42·71-s − 0.468·73-s − 0.455·77-s − 0.562·79-s + 81-s − 1.86·83-s − 0.317·89-s + 0.104·91-s − 1.31·97-s − 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5811583886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5811583886\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−14.38933733901150, −13.70783164265152, −13.23003801545497, −12.66334816988387, −12.34293618444404, −11.57136452375657, −11.20897271076771, −10.87978765045784, −10.15528416857596, −9.473563581421471, −9.142954069125433, −8.619842459232673, −8.217651416148607, −7.387644302516885, −6.899269614933307, −6.311716842228716, −5.911388938290068, −5.379368078551393, −4.414482011526006, −4.087063424167040, −3.504495506617300, −2.621491984117795, −2.179728714305128, −1.389214028898688, −0.2500944427676432,
0.2500944427676432, 1.389214028898688, 2.179728714305128, 2.621491984117795, 3.504495506617300, 4.087063424167040, 4.414482011526006, 5.379368078551393, 5.911388938290068, 6.311716842228716, 6.899269614933307, 7.387644302516885, 8.217651416148607, 8.619842459232673, 9.142954069125433, 9.473563581421471, 10.15528416857596, 10.87978765045784, 11.20897271076771, 11.57136452375657, 12.34293618444404, 12.66334816988387, 13.23003801545497, 13.70783164265152, 14.38933733901150
