| L(s) = 1 | + 2.98·3-s + 1.64·5-s + 7-s + 5.89·9-s + 11-s + 13-s + 4.91·15-s + 6.79·17-s − 4.90·19-s + 2.98·21-s + 3.96·23-s − 2.28·25-s + 8.62·27-s − 1.97·29-s + 9.49·31-s + 2.98·33-s + 1.64·35-s − 10.6·37-s + 2.98·39-s − 0.717·41-s + 6.18·43-s + 9.71·45-s − 12.0·47-s + 49-s + 20.2·51-s + 12.1·53-s + 1.64·55-s + ⋯ |
| L(s) = 1 | + 1.72·3-s + 0.737·5-s + 0.377·7-s + 1.96·9-s + 0.301·11-s + 0.277·13-s + 1.26·15-s + 1.64·17-s − 1.12·19-s + 0.650·21-s + 0.826·23-s − 0.456·25-s + 1.65·27-s − 0.367·29-s + 1.70·31-s + 0.519·33-s + 0.278·35-s − 1.75·37-s + 0.477·39-s − 0.112·41-s + 0.943·43-s + 1.44·45-s − 1.75·47-s + 0.142·49-s + 2.83·51-s + 1.67·53-s + 0.222·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.575078395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.575078395\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 2.98T + 3T^{2} \) |
| 5 | \( 1 - 1.64T + 5T^{2} \) |
| 17 | \( 1 - 6.79T + 17T^{2} \) |
| 19 | \( 1 + 4.90T + 19T^{2} \) |
| 23 | \( 1 - 3.96T + 23T^{2} \) |
| 29 | \( 1 + 1.97T + 29T^{2} \) |
| 31 | \( 1 - 9.49T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + 0.717T + 41T^{2} \) |
| 43 | \( 1 - 6.18T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 + 2.55T + 59T^{2} \) |
| 61 | \( 1 - 2.51T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 6.36T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 + 4.62T + 83T^{2} \) |
| 89 | \( 1 + 0.496T + 89T^{2} \) |
| 97 | \( 1 + 4.84T + 97T^{2} \) |
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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
−8.104677592798991054382462691018, −7.28784759311730259760448402053, −6.61511419214452976560578664786, −5.76377846370417287138435203246, −4.93841383368281285382798074506, −4.05345762382625489666147539855, −3.39845870663694622719280150727, −2.67395066277771107415850946308, −1.88403358409488244793896738238, −1.20331921951806049551680453513,
1.20331921951806049551680453513, 1.88403358409488244793896738238, 2.67395066277771107415850946308, 3.39845870663694622719280150727, 4.05345762382625489666147539855, 4.93841383368281285382798074506, 5.76377846370417287138435203246, 6.61511419214452976560578664786, 7.28784759311730259760448402053, 8.104677592798991054382462691018
