| L(s) = 1 | − 2-s − 3-s + 4-s − 3.41·5-s + 6-s − 8-s + 9-s + 3.41·10-s − 4.73·11-s − 12-s + 13-s + 3.41·15-s + 16-s + 7.79·17-s − 18-s + 2.68·19-s − 3.41·20-s + 4.73·22-s − 3.73·23-s + 24-s + 6.68·25-s − 26-s − 27-s − 3·29-s − 3.41·30-s − 5.68·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.52·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 1.08·10-s − 1.42·11-s − 0.288·12-s + 0.277·13-s + 0.882·15-s + 0.250·16-s + 1.89·17-s − 0.235·18-s + 0.616·19-s − 0.764·20-s + 1.00·22-s − 0.777·23-s + 0.204·24-s + 1.33·25-s − 0.196·26-s − 0.192·27-s − 0.557·29-s − 0.624·30-s − 1.02·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3827767471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3827767471\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 3.41T + 5T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 17 | \( 1 - 7.79T + 17T^{2} \) |
| 19 | \( 1 - 2.68T + 19T^{2} \) |
| 23 | \( 1 + 3.73T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 5.68T + 31T^{2} \) |
| 37 | \( 1 + 3.10T + 37T^{2} \) |
| 41 | \( 1 - 0.892T + 41T^{2} \) |
| 43 | \( 1 + 12.8T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 9.21T + 53T^{2} \) |
| 59 | \( 1 + 6.10T + 59T^{2} \) |
| 61 | \( 1 - 3.04T + 61T^{2} \) |
| 67 | \( 1 + 0.0652T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 0.934T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 + 4.79T + 97T^{2} \) |
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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
−8.069321157868625732995408011999, −7.893308461358242503114372487445, −7.37965767220777498487660087177, −6.40431647299657545263415584140, −5.42926046519505342525594085725, −4.90464322594887548280334193632, −3.59268125934152572543779077269, −3.21407782880034820566940973282, −1.68183866631211340206989375717, −0.40877548670727699645444367385,
0.40877548670727699645444367385, 1.68183866631211340206989375717, 3.21407782880034820566940973282, 3.59268125934152572543779077269, 4.90464322594887548280334193632, 5.42926046519505342525594085725, 6.40431647299657545263415584140, 7.37965767220777498487660087177, 7.893308461358242503114372487445, 8.069321157868625732995408011999
