| L(s) = 1 | + 3-s + 1.86·5-s + 4.94·7-s + 9-s − 2.50·11-s + 4.49·13-s + 1.86·15-s − 4.49·19-s + 4.94·21-s + 4.18·23-s − 1.50·25-s + 27-s − 4.65·29-s + 1.40·31-s − 2.50·33-s + 9.24·35-s − 8.70·37-s + 4.49·39-s + 5.73·41-s + 2.50·43-s + 1.86·45-s − 1.33·47-s + 17.4·49-s + 8.18·53-s − 4.68·55-s − 4.49·57-s + 5.55·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.836·5-s + 1.86·7-s + 0.333·9-s − 0.755·11-s + 1.24·13-s + 0.482·15-s − 1.03·19-s + 1.07·21-s + 0.873·23-s − 0.300·25-s + 0.192·27-s − 0.864·29-s + 0.252·31-s − 0.435·33-s + 1.56·35-s − 1.43·37-s + 0.718·39-s + 0.895·41-s + 0.381·43-s + 0.278·45-s − 0.195·47-s + 2.49·49-s + 1.12·53-s − 0.631·55-s − 0.594·57-s + 0.723·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.053418138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.053418138\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 1.86T + 5T^{2} \) |
| 7 | \( 1 - 4.94T + 7T^{2} \) |
| 11 | \( 1 + 2.50T + 11T^{2} \) |
| 13 | \( 1 - 4.49T + 13T^{2} \) |
| 19 | \( 1 + 4.49T + 19T^{2} \) |
| 23 | \( 1 - 4.18T + 23T^{2} \) |
| 29 | \( 1 + 4.65T + 29T^{2} \) |
| 31 | \( 1 - 1.40T + 31T^{2} \) |
| 37 | \( 1 + 8.70T + 37T^{2} \) |
| 41 | \( 1 - 5.73T + 41T^{2} \) |
| 43 | \( 1 - 2.50T + 43T^{2} \) |
| 47 | \( 1 + 1.33T + 47T^{2} \) |
| 53 | \( 1 - 8.18T + 53T^{2} \) |
| 59 | \( 1 - 5.55T + 59T^{2} \) |
| 61 | \( 1 - 9.66T + 61T^{2} \) |
| 67 | \( 1 - 2.99T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + 8.67T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 + 2.63T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−8.050421855731855681271965625580, −7.43600959409032205472568370310, −6.59405438400397193733042374004, −5.62627617888576503926287784912, −5.24655282776630415789492514804, −4.35462121520353713874223165811, −3.65219731687093716103889863395, −2.40751372896520092759122815544, −1.93865824376808817792853922655, −1.08339449637689354649922284454,
1.08339449637689354649922284454, 1.93865824376808817792853922655, 2.40751372896520092759122815544, 3.65219731687093716103889863395, 4.35462121520353713874223165811, 5.24655282776630415789492514804, 5.62627617888576503926287784912, 6.59405438400397193733042374004, 7.43600959409032205472568370310, 8.050421855731855681271965625580
