| L(s) = 1 | − 3-s − 5-s − 0.941·7-s + 9-s + 1.81·11-s − 5.81·13-s + 15-s + 2.76·17-s + 0.102·19-s + 0.941·21-s + 0.798·23-s + 25-s − 27-s − 3.73·29-s + 8.68·31-s − 1.81·33-s + 0.941·35-s − 4.68·37-s + 5.81·39-s − 2.78·41-s − 9.62·43-s − 45-s − 6.94·47-s − 6.11·49-s − 2.76·51-s + 4.87·53-s − 1.81·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.355·7-s + 0.333·9-s + 0.545·11-s − 1.61·13-s + 0.258·15-s + 0.671·17-s + 0.0234·19-s + 0.205·21-s + 0.166·23-s + 0.200·25-s − 0.192·27-s − 0.694·29-s + 1.56·31-s − 0.315·33-s + 0.159·35-s − 0.769·37-s + 0.931·39-s − 0.434·41-s − 1.46·43-s − 0.149·45-s − 1.01·47-s − 0.873·49-s − 0.387·51-s + 0.669·53-s − 0.244·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.001119255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.001119255\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 + 0.941T + 7T^{2} \) |
| 11 | \( 1 - 1.81T + 11T^{2} \) |
| 13 | \( 1 + 5.81T + 13T^{2} \) |
| 17 | \( 1 - 2.76T + 17T^{2} \) |
| 19 | \( 1 - 0.102T + 19T^{2} \) |
| 23 | \( 1 - 0.798T + 23T^{2} \) |
| 29 | \( 1 + 3.73T + 29T^{2} \) |
| 31 | \( 1 - 8.68T + 31T^{2} \) |
| 37 | \( 1 + 4.68T + 37T^{2} \) |
| 41 | \( 1 + 2.78T + 41T^{2} \) |
| 43 | \( 1 + 9.62T + 43T^{2} \) |
| 47 | \( 1 + 6.94T + 47T^{2} \) |
| 53 | \( 1 - 4.87T + 53T^{2} \) |
| 59 | \( 1 + 2.95T + 59T^{2} \) |
| 61 | \( 1 - 8.87T + 61T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 2.15T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 + 2.16T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−8.293884933945480305504176779294, −7.69233032736424471165249600872, −6.81973625899560127407179432050, −6.46637576946380244447825241237, −5.24527341611261590106843560173, −4.89284869638914594322231834259, −3.84741834992210902815834000185, −3.06264264788907232673074317661, −1.90696024646604125346164910534, −0.57926817496702664226590570825,
0.57926817496702664226590570825, 1.90696024646604125346164910534, 3.06264264788907232673074317661, 3.84741834992210902815834000185, 4.89284869638914594322231834259, 5.24527341611261590106843560173, 6.46637576946380244447825241237, 6.81973625899560127407179432050, 7.69233032736424471165249600872, 8.293884933945480305504176779294
