| L(s) = 1 | − 1.50·2-s − 5.73·4-s − 18.0·5-s − 23.9·7-s + 20.6·8-s + 27.2·10-s + 55.0·11-s − 40.0·13-s + 35.9·14-s + 14.7·16-s − 46.7·17-s + 116.·19-s + 103.·20-s − 82.8·22-s − 131.·23-s + 201.·25-s + 60.2·26-s + 137.·28-s − 268.·29-s − 82.3·31-s − 187.·32-s + 70.4·34-s + 432.·35-s − 208.·37-s − 175.·38-s − 373.·40-s + 112.·41-s + ⋯ |
| L(s) = 1 | − 0.532·2-s − 0.716·4-s − 1.61·5-s − 1.29·7-s + 0.913·8-s + 0.860·10-s + 1.50·11-s − 0.854·13-s + 0.686·14-s + 0.230·16-s − 0.667·17-s + 1.40·19-s + 1.15·20-s − 0.803·22-s − 1.19·23-s + 1.61·25-s + 0.454·26-s + 0.925·28-s − 1.72·29-s − 0.476·31-s − 1.03·32-s + 0.355·34-s + 2.08·35-s − 0.926·37-s − 0.748·38-s − 1.47·40-s + 0.427·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.008505511643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.008505511643\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 + 1.50T + 8T^{2} \) |
| 5 | \( 1 + 18.0T + 125T^{2} \) |
| 7 | \( 1 + 23.9T + 343T^{2} \) |
| 11 | \( 1 - 55.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 40.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 46.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 131.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 268.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 82.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 208.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 112.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 14.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 509.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 58.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 272.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 102.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 203.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 791.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 807.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 211.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 251.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.15e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 422.T + 9.12e5T^{2} \) |
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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
−8.889790283949332864251908654539, −7.910640906383749406203043819611, −7.33260001640551355111839081488, −6.69501877583507319641756213177, −5.52603509694640807457750184661, −4.38098539079021888849368714520, −3.84130494768077176589710200348, −3.20177724469319619476771915246, −1.46024666740895055993043876092, −0.04780773724181678181415320957,
0.04780773724181678181415320957, 1.46024666740895055993043876092, 3.20177724469319619476771915246, 3.84130494768077176589710200348, 4.38098539079021888849368714520, 5.52603509694640807457750184661, 6.69501877583507319641756213177, 7.33260001640551355111839081488, 7.910640906383749406203043819611, 8.889790283949332864251908654539
