| L(s) = 1 | − 2.34·3-s − 11.5·5-s − 8.54·7-s − 21.5·9-s + 47.4·11-s − 81.9·13-s + 27.0·15-s − 49.9·17-s − 20.4·19-s + 20.0·21-s − 105.·23-s + 8.57·25-s + 113.·27-s − 29·29-s − 22.4·31-s − 111.·33-s + 98.7·35-s − 316.·37-s + 192.·39-s − 427.·41-s − 190.·43-s + 248.·45-s − 459.·47-s − 270.·49-s + 117.·51-s + 545.·53-s − 548.·55-s + ⋯ |
| L(s) = 1 | − 0.451·3-s − 1.03·5-s − 0.461·7-s − 0.796·9-s + 1.30·11-s − 1.74·13-s + 0.466·15-s − 0.712·17-s − 0.246·19-s + 0.208·21-s − 0.954·23-s + 0.0685·25-s + 0.810·27-s − 0.185·29-s − 0.130·31-s − 0.587·33-s + 0.476·35-s − 1.40·37-s + 0.788·39-s − 1.62·41-s − 0.676·43-s + 0.823·45-s − 1.42·47-s − 0.787·49-s + 0.321·51-s + 1.41·53-s − 1.34·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.04796351166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04796351166\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 + 2.34T + 27T^{2} \) |
| 5 | \( 1 + 11.5T + 125T^{2} \) |
| 7 | \( 1 + 8.54T + 343T^{2} \) |
| 11 | \( 1 - 47.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 81.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 49.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 20.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 105.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 22.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 316.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 427.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 190.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 459.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 545.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 809.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 67.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 822.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 666.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 275.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 681.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.07e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 229.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 261.T + 9.12e5T^{2} \) |
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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
−8.761681560486049207249352507000, −8.198638590916045392655001881027, −7.06359492531593800729899475652, −6.72338575874362184863070910781, −5.65536311386665703683022615777, −4.74673805567748201249597162091, −3.93446503318803640802880007490, −3.06512599842420614835951780540, −1.84559128665715248508101841910, −0.099321308691437306871733765022,
0.099321308691437306871733765022, 1.84559128665715248508101841910, 3.06512599842420614835951780540, 3.93446503318803640802880007490, 4.74673805567748201249597162091, 5.65536311386665703683022615777, 6.72338575874362184863070910781, 7.06359492531593800729899475652, 8.198638590916045392655001881027, 8.761681560486049207249352507000
