| L(s) = 1 | − 1.23·5-s − 3.23·7-s + 4·11-s − 4.47·13-s + 2.76·17-s − 7.23·19-s + 23-s − 3.47·25-s − 4.47·29-s + 6.47·31-s + 4.00·35-s + 4.47·37-s + 10.9·41-s + 5.70·43-s − 4·47-s + 3.47·49-s + 5.23·53-s − 4.94·55-s − 4.94·59-s + 4.47·61-s + 5.52·65-s − 0.763·67-s − 8·71-s + 6.94·73-s − 12.9·77-s − 9.70·79-s + 4·83-s + ⋯ |
| L(s) = 1 | − 0.552·5-s − 1.22·7-s + 1.20·11-s − 1.24·13-s + 0.670·17-s − 1.66·19-s + 0.208·23-s − 0.694·25-s − 0.830·29-s + 1.16·31-s + 0.676·35-s + 0.735·37-s + 1.70·41-s + 0.870·43-s − 0.583·47-s + 0.496·49-s + 0.719·53-s − 0.666·55-s − 0.643·59-s + 0.572·61-s + 0.685·65-s − 0.0933·67-s − 0.949·71-s + 0.812·73-s − 1.47·77-s − 1.09·79-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.091295310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.091295310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 + 3.23T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 2.76T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 5.70T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 5.23T + 53T^{2} \) |
| 59 | \( 1 + 4.94T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 + 0.763T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 6.94T + 73T^{2} \) |
| 79 | \( 1 + 9.70T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 1.23T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
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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
−8.712891157107573231341201107392, −7.77649927735117120452805303660, −7.16088565261667160671623349078, −6.35306243702147408864019493251, −5.85979828755834596311599872827, −4.52420756293419912041675880639, −4.01174770139087270036621913846, −3.10438017349863339973616654813, −2.15425612428814953107777231397, −0.60094132722387482727424086384,
0.60094132722387482727424086384, 2.15425612428814953107777231397, 3.10438017349863339973616654813, 4.01174770139087270036621913846, 4.52420756293419912041675880639, 5.85979828755834596311599872827, 6.35306243702147408864019493251, 7.16088565261667160671623349078, 7.77649927735117120452805303660, 8.712891157107573231341201107392
