| L(s) = 1 | − 6.28·3-s − 18.9·5-s + 32.3·7-s + 12.4·9-s + 29.5·11-s + 0.0104·13-s + 119.·15-s − 9.38·17-s + 98.7·19-s − 202.·21-s + 152.·23-s + 235.·25-s + 91.4·27-s − 29·29-s + 192.·31-s − 185.·33-s − 613.·35-s − 3.01·37-s − 0.0653·39-s − 140.·41-s − 346.·43-s − 236.·45-s − 282.·47-s + 700.·49-s + 58.9·51-s − 628.·53-s − 561.·55-s + ⋯ |
| L(s) = 1 | − 1.20·3-s − 1.69·5-s + 1.74·7-s + 0.460·9-s + 0.809·11-s + 0.000221·13-s + 2.05·15-s − 0.133·17-s + 1.19·19-s − 2.10·21-s + 1.38·23-s + 1.88·25-s + 0.651·27-s − 0.185·29-s + 1.11·31-s − 0.978·33-s − 2.96·35-s − 0.0134·37-s − 0.000268·39-s − 0.534·41-s − 1.22·43-s − 0.783·45-s − 0.875·47-s + 2.04·49-s + 0.161·51-s − 1.62·53-s − 1.37·55-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
| \(L(2)\) |
\(\approx\) |
\(1.297222365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.297222365\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 6.28T + 27T^{2} \) |
| 5 | \( 1 + 18.9T + 125T^{2} \) |
| 7 | \( 1 - 32.3T + 343T^{2} \) |
| 11 | \( 1 - 29.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 0.0104T + 2.19e3T^{2} \) |
| 17 | \( 1 + 9.38T + 4.91e3T^{2} \) |
| 19 | \( 1 - 98.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 152.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 192.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 3.01T + 5.06e4T^{2} \) |
| 41 | \( 1 + 140.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 346.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 282.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 628.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 586.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 818.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 463.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 23.3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 83.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 141.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 994.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 270.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.23e3T + 9.12e5T^{2} \) |
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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
−8.456364955579919814076275433971, −8.227133265071480628214249277810, −7.20139923777324925194456709637, −6.70745082906611919401718701810, −5.33564919672808044975940360405, −4.88948489516090838858881092021, −4.18134938896574318978557443114, −3.14117668670011759425667501309, −1.37959430839094073395948660818, −0.64825932977477679299733350988,
0.64825932977477679299733350988, 1.37959430839094073395948660818, 3.14117668670011759425667501309, 4.18134938896574318978557443114, 4.88948489516090838858881092021, 5.33564919672808044975940360405, 6.70745082906611919401718701810, 7.20139923777324925194456709637, 8.227133265071480628214249277810, 8.456364955579919814076275433971
