| L(s) = 1 | − 1.50·3-s − 2.88·5-s − 1.88·7-s − 0.748·9-s + 11-s − 1.08·13-s + 4.32·15-s − 4.80·17-s − 19-s + 2.82·21-s − 8.51·23-s + 3.31·25-s + 5.62·27-s − 7.97·29-s + 4.32·31-s − 1.50·33-s + 5.43·35-s − 7.21·37-s + 1.62·39-s − 8.45·41-s − 3.83·43-s + 2.15·45-s − 4.50·47-s − 3.45·49-s + 7.20·51-s − 6.88·53-s − 2.88·55-s + ⋯ |
| L(s) = 1 | − 0.866·3-s − 1.28·5-s − 0.712·7-s − 0.249·9-s + 0.301·11-s − 0.300·13-s + 1.11·15-s − 1.16·17-s − 0.229·19-s + 0.616·21-s − 1.77·23-s + 0.663·25-s + 1.08·27-s − 1.48·29-s + 0.777·31-s − 0.261·33-s + 0.918·35-s − 1.18·37-s + 0.260·39-s − 1.32·41-s − 0.584·43-s + 0.321·45-s − 0.657·47-s − 0.492·49-s + 1.00·51-s − 0.945·53-s − 0.388·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09716182193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09716182193\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.50T + 3T^{2} \) |
| 5 | \( 1 + 2.88T + 5T^{2} \) |
| 7 | \( 1 + 1.88T + 7T^{2} \) |
| 13 | \( 1 + 1.08T + 13T^{2} \) |
| 17 | \( 1 + 4.80T + 17T^{2} \) |
| 23 | \( 1 + 8.51T + 23T^{2} \) |
| 29 | \( 1 + 7.97T + 29T^{2} \) |
| 31 | \( 1 - 4.32T + 31T^{2} \) |
| 37 | \( 1 + 7.21T + 37T^{2} \) |
| 41 | \( 1 + 8.45T + 41T^{2} \) |
| 43 | \( 1 + 3.83T + 43T^{2} \) |
| 47 | \( 1 + 4.50T + 47T^{2} \) |
| 53 | \( 1 + 6.88T + 53T^{2} \) |
| 59 | \( 1 + 1.50T + 59T^{2} \) |
| 61 | \( 1 - 3.50T + 61T^{2} \) |
| 67 | \( 1 - 7.29T + 67T^{2} \) |
| 71 | \( 1 - 7.56T + 71T^{2} \) |
| 73 | \( 1 - 9.30T + 73T^{2} \) |
| 79 | \( 1 + 0.234T + 79T^{2} \) |
| 83 | \( 1 - 4.64T + 83T^{2} \) |
| 89 | \( 1 + 7.24T + 89T^{2} \) |
| 97 | \( 1 + 8.05T + 97T^{2} \) |
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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
−8.391646150739593565285219871109, −8.019595334043364047822999859328, −6.83936631243810056705989698823, −6.57490850168713015571794885783, −5.62216160497407068015130401667, −4.76590993844041293512920215441, −3.95043851663601243894290011628, −3.28722071276850797779382956411, −1.98297966392096619703219094295, −0.18199942745580164886373080749,
0.18199942745580164886373080749, 1.98297966392096619703219094295, 3.28722071276850797779382956411, 3.95043851663601243894290011628, 4.76590993844041293512920215441, 5.62216160497407068015130401667, 6.57490850168713015571794885783, 6.83936631243810056705989698823, 8.019595334043364047822999859328, 8.391646150739593565285219871109
