| L(s) = 1 | + 0.753·2-s − 0.885·3-s − 1.43·4-s − 0.666·6-s + 7-s − 2.58·8-s − 2.21·9-s + 2.65·11-s + 1.26·12-s + 0.916·13-s + 0.753·14-s + 0.918·16-s + 1.58·17-s − 1.66·18-s + 1.01·19-s − 0.885·21-s + 1.99·22-s + 7.15·23-s + 2.28·24-s + 0.690·26-s + 4.61·27-s − 1.43·28-s − 0.550·29-s + 0.295·31-s + 5.86·32-s − 2.34·33-s + 1.19·34-s + ⋯ |
| L(s) = 1 | + 0.532·2-s − 0.511·3-s − 0.716·4-s − 0.272·6-s + 0.377·7-s − 0.914·8-s − 0.738·9-s + 0.799·11-s + 0.366·12-s + 0.254·13-s + 0.201·14-s + 0.229·16-s + 0.384·17-s − 0.393·18-s + 0.233·19-s − 0.193·21-s + 0.425·22-s + 1.49·23-s + 0.467·24-s + 0.135·26-s + 0.888·27-s − 0.270·28-s − 0.102·29-s + 0.0529·31-s + 1.03·32-s − 0.408·33-s + 0.204·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.395127033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.395127033\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - 0.753T + 2T^{2} \) |
| 3 | \( 1 + 0.885T + 3T^{2} \) |
| 11 | \( 1 - 2.65T + 11T^{2} \) |
| 13 | \( 1 - 0.916T + 13T^{2} \) |
| 17 | \( 1 - 1.58T + 17T^{2} \) |
| 19 | \( 1 - 1.01T + 19T^{2} \) |
| 23 | \( 1 - 7.15T + 23T^{2} \) |
| 29 | \( 1 + 0.550T + 29T^{2} \) |
| 31 | \( 1 - 0.295T + 31T^{2} \) |
| 37 | \( 1 - 5.09T + 37T^{2} \) |
| 41 | \( 1 + 5.01T + 41T^{2} \) |
| 43 | \( 1 - 5.09T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 1.30T + 53T^{2} \) |
| 59 | \( 1 - 4.32T + 59T^{2} \) |
| 61 | \( 1 - 2.78T + 61T^{2} \) |
| 67 | \( 1 + 7.59T + 67T^{2} \) |
| 71 | \( 1 + 8.70T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 + 3.51T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−10.20805358625892457357502115745, −9.091641732745179641268833707154, −8.727113655596282211996328497039, −7.56752892774227223292549979071, −6.41079699426416548103566077391, −5.62555498579545660464847209765, −4.90687800399370423865555418038, −3.93125573874472580107989503683, −2.88462179164816194604585329242, −0.937385988968963932118313623538,
0.937385988968963932118313623538, 2.88462179164816194604585329242, 3.93125573874472580107989503683, 4.90687800399370423865555418038, 5.62555498579545660464847209765, 6.41079699426416548103566077391, 7.56752892774227223292549979071, 8.727113655596282211996328497039, 9.091641732745179641268833707154, 10.20805358625892457357502115745
