L(s) = 1 | + 1.44·2-s − 3-s + 0.0877·4-s + 4.08·5-s − 1.44·6-s − 1.49·7-s − 2.76·8-s + 9-s + 5.90·10-s + 1.31·11-s − 0.0877·12-s − 13-s − 2.16·14-s − 4.08·15-s − 4.16·16-s − 6.35·17-s + 1.44·18-s + 1.16·19-s + 0.358·20-s + 1.49·21-s + 1.90·22-s − 7.75·23-s + 2.76·24-s + 11.7·25-s − 1.44·26-s − 27-s − 0.131·28-s + ⋯ |
L(s) = 1 | + 1.02·2-s − 0.577·3-s + 0.0438·4-s + 1.82·5-s − 0.589·6-s − 0.565·7-s − 0.976·8-s + 0.333·9-s + 1.86·10-s + 0.397·11-s − 0.0253·12-s − 0.277·13-s − 0.577·14-s − 1.05·15-s − 1.04·16-s − 1.54·17-s + 0.340·18-s + 0.266·19-s + 0.0801·20-s + 0.326·21-s + 0.405·22-s − 1.61·23-s + 0.564·24-s + 2.34·25-s − 0.283·26-s − 0.192·27-s − 0.0247·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 - 1.44T + 2T^{2} \) |
| 5 | \( 1 - 4.08T + 5T^{2} \) |
| 7 | \( 1 + 1.49T + 7T^{2} \) |
| 11 | \( 1 - 1.31T + 11T^{2} \) |
| 17 | \( 1 + 6.35T + 17T^{2} \) |
| 19 | \( 1 - 1.16T + 19T^{2} \) |
| 23 | \( 1 + 7.75T + 23T^{2} \) |
| 29 | \( 1 + 9.19T + 29T^{2} \) |
| 31 | \( 1 + 0.815T + 31T^{2} \) |
| 37 | \( 1 - 7.09T + 37T^{2} \) |
| 41 | \( 1 - 1.50T + 41T^{2} \) |
| 43 | \( 1 + 3.76T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 1.59T + 59T^{2} \) |
| 61 | \( 1 - 2.44T + 61T^{2} \) |
| 67 | \( 1 - 3.70T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 - 4.74T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 + 7.11T + 89T^{2} \) |
| 97 | \( 1 + 9.99T + 97T^{2} \) |
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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
−8.082592330026732670317506785768, −6.67466106339924473878999125377, −6.44323291396660737617618979016, −5.80428295099530239971584340195, −5.18744256050603771213295313276, −4.45343688873136393582167317703, −3.56531449649742474194010441145, −2.47027303792669064190080340372, −1.74412279372869300105861416074, 0,
1.74412279372869300105861416074, 2.47027303792669064190080340372, 3.56531449649742474194010441145, 4.45343688873136393582167317703, 5.18744256050603771213295313276, 5.80428295099530239971584340195, 6.44323291396660737617618979016, 6.67466106339924473878999125377, 8.082592330026732670317506785768