| L(s) = 1 | − 2.60·2-s + 3-s + 4.79·4-s − 5-s − 2.60·6-s − 3.84·7-s − 7.27·8-s + 9-s + 2.60·10-s + 1.93·11-s + 4.79·12-s − 2.50·13-s + 10.0·14-s − 15-s + 9.38·16-s − 5.96·17-s − 2.60·18-s + 0.0654·19-s − 4.79·20-s − 3.84·21-s − 5.04·22-s + 8.51·23-s − 7.27·24-s + 25-s + 6.53·26-s + 27-s − 18.4·28-s + ⋯ |
| L(s) = 1 | − 1.84·2-s + 0.577·3-s + 2.39·4-s − 0.447·5-s − 1.06·6-s − 1.45·7-s − 2.57·8-s + 0.333·9-s + 0.824·10-s + 0.583·11-s + 1.38·12-s − 0.695·13-s + 2.67·14-s − 0.258·15-s + 2.34·16-s − 1.44·17-s − 0.614·18-s + 0.0150·19-s − 1.07·20-s − 0.838·21-s − 1.07·22-s + 1.77·23-s − 1.48·24-s + 0.200·25-s + 1.28·26-s + 0.192·27-s − 3.48·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 7 | \( 1 + 3.84T + 7T^{2} \) |
| 11 | \( 1 - 1.93T + 11T^{2} \) |
| 13 | \( 1 + 2.50T + 13T^{2} \) |
| 17 | \( 1 + 5.96T + 17T^{2} \) |
| 19 | \( 1 - 0.0654T + 19T^{2} \) |
| 23 | \( 1 - 8.51T + 23T^{2} \) |
| 29 | \( 1 + 1.45T + 29T^{2} \) |
| 31 | \( 1 + 1.78T + 31T^{2} \) |
| 37 | \( 1 - 3.57T + 37T^{2} \) |
| 41 | \( 1 - 9.75T + 41T^{2} \) |
| 43 | \( 1 - 0.475T + 43T^{2} \) |
| 47 | \( 1 - 1.15T + 47T^{2} \) |
| 53 | \( 1 - 0.959T + 53T^{2} \) |
| 59 | \( 1 + 5.91T + 59T^{2} \) |
| 61 | \( 1 + 5.37T + 61T^{2} \) |
| 67 | \( 1 + 9.36T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 + 7.18T + 73T^{2} \) |
| 79 | \( 1 - 2.26T + 79T^{2} \) |
| 83 | \( 1 - 2.50T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−7.60848573646541322235357926015, −7.38458960367946012466548307588, −6.57384285249451191108977379379, −6.21897474080279655364273026996, −4.74607157387059225519431339293, −3.67480194769082493932505089562, −2.85596242763574046060776272439, −2.26787671837702149387776475781, −1.00000798956009390353594789288, 0,
1.00000798956009390353594789288, 2.26787671837702149387776475781, 2.85596242763574046060776272439, 3.67480194769082493932505089562, 4.74607157387059225519431339293, 6.21897474080279655364273026996, 6.57384285249451191108977379379, 7.38458960367946012466548307588, 7.60848573646541322235357926015
