| L(s) = 1 | + 2.23·2-s − 3-s + 3.00·4-s + 1.61·5-s − 2.23·6-s − 7-s + 2.23·8-s + 9-s + 3.61·10-s − 3.00·12-s + 1.23·13-s − 2.23·14-s − 1.61·15-s − 0.999·16-s + 1.85·17-s + 2.23·18-s + 3.85·19-s + 4.85·20-s + 21-s + 6.61·23-s − 2.23·24-s − 2.38·25-s + 2.76·26-s − 27-s − 3.00·28-s + 6·29-s − 3.61·30-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 0.577·3-s + 1.50·4-s + 0.723·5-s − 0.912·6-s − 0.377·7-s + 0.790·8-s + 0.333·9-s + 1.14·10-s − 0.866·12-s + 0.342·13-s − 0.597·14-s − 0.417·15-s − 0.249·16-s + 0.449·17-s + 0.527·18-s + 0.884·19-s + 1.08·20-s + 0.218·21-s + 1.37·23-s − 0.456·24-s − 0.476·25-s + 0.542·26-s − 0.192·27-s − 0.566·28-s + 1.11·29-s − 0.660·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.313229114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.313229114\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 1.85T + 17T^{2} \) |
| 19 | \( 1 - 3.85T + 19T^{2} \) |
| 23 | \( 1 - 6.61T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 8.09T + 31T^{2} \) |
| 37 | \( 1 - 2.38T + 37T^{2} \) |
| 41 | \( 1 + 9.61T + 41T^{2} \) |
| 43 | \( 1 - 3.70T + 43T^{2} \) |
| 47 | \( 1 + 4.47T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 1.23T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + 0.291T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 7.52T + 83T^{2} \) |
| 89 | \( 1 - 3.38T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.023743568221924504216409428378, −7.914415105534707878010448230928, −6.72716092899351277528219788226, −6.51921128365278261715880817369, −5.50879013082944547998349927684, −5.18154083423738583247014252208, −4.23853109921884502856414249422, −3.28835259961409353657930448881, −2.53682875350577112771699235860, −1.15114229669994717867736904528,
1.15114229669994717867736904528, 2.53682875350577112771699235860, 3.28835259961409353657930448881, 4.23853109921884502856414249422, 5.18154083423738583247014252208, 5.50879013082944547998349927684, 6.51921128365278261715880817369, 6.72716092899351277528219788226, 7.914415105534707878010448230928, 9.023743568221924504216409428378
