| L(s) = 1 | + 5-s − 1.88·7-s − 1.03·11-s + 4.53·13-s − 0.816·17-s − 6.23·19-s + 0.616·23-s + 25-s + 2.43·29-s − 0.383·31-s − 1.88·35-s − 5.88·37-s − 0.0991·41-s + 7.34·43-s − 6.89·47-s − 3.44·49-s − 8.95·53-s − 1.03·55-s − 1.73·59-s + 10.7·61-s + 4.53·65-s − 12.8·67-s + 8.96·71-s + 6.41·73-s + 1.94·77-s − 0.535·79-s − 8.04·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.712·7-s − 0.310·11-s + 1.25·13-s − 0.198·17-s − 1.43·19-s + 0.128·23-s + 0.200·25-s + 0.451·29-s − 0.0688·31-s − 0.318·35-s − 0.966·37-s − 0.0154·41-s + 1.12·43-s − 1.00·47-s − 0.492·49-s − 1.23·53-s − 0.139·55-s − 0.225·59-s + 1.37·61-s + 0.562·65-s − 1.57·67-s + 1.06·71-s + 0.751·73-s + 0.221·77-s − 0.0602·79-s − 0.883·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 1.88T + 7T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 13 | \( 1 - 4.53T + 13T^{2} \) |
| 17 | \( 1 + 0.816T + 17T^{2} \) |
| 19 | \( 1 + 6.23T + 19T^{2} \) |
| 23 | \( 1 - 0.616T + 23T^{2} \) |
| 29 | \( 1 - 2.43T + 29T^{2} \) |
| 31 | \( 1 + 0.383T + 31T^{2} \) |
| 37 | \( 1 + 5.88T + 37T^{2} \) |
| 41 | \( 1 + 0.0991T + 41T^{2} \) |
| 43 | \( 1 - 7.34T + 43T^{2} \) |
| 47 | \( 1 + 6.89T + 47T^{2} \) |
| 53 | \( 1 + 8.95T + 53T^{2} \) |
| 59 | \( 1 + 1.73T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 8.96T + 71T^{2} \) |
| 73 | \( 1 - 6.41T + 73T^{2} \) |
| 79 | \( 1 + 0.535T + 79T^{2} \) |
| 83 | \( 1 + 8.04T + 83T^{2} \) |
| 89 | \( 1 + 18.6T + 89T^{2} \) |
| 97 | \( 1 - 7.51T + 97T^{2} \) |
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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
−7.70563221736632188324070947836, −6.66908831488873099700285734798, −6.38405162835866349952069294343, −5.67021972017690320419486452180, −4.78682245891604885499481735250, −3.94576574135476064578232111294, −3.20121127914915394671140853899, −2.31621785707377617863700511445, −1.35219283967426282493437666435, 0,
1.35219283967426282493437666435, 2.31621785707377617863700511445, 3.20121127914915394671140853899, 3.94576574135476064578232111294, 4.78682245891604885499481735250, 5.67021972017690320419486452180, 6.38405162835866349952069294343, 6.66908831488873099700285734798, 7.70563221736632188324070947836
