L(s) = 1 | − 1.86·2-s + 2.64·3-s + 1.49·4-s − 2.58·5-s − 4.95·6-s + 0.553·7-s + 0.941·8-s + 4.00·9-s + 4.82·10-s + 5.54·11-s + 3.96·12-s + 2.13·13-s − 1.03·14-s − 6.83·15-s − 4.75·16-s + 17-s − 7.49·18-s − 5.27·19-s − 3.86·20-s + 1.46·21-s − 10.3·22-s − 2.77·23-s + 2.49·24-s + 1.67·25-s − 3.99·26-s + 2.66·27-s + 0.827·28-s + ⋯ |
L(s) = 1 | − 1.32·2-s + 1.52·3-s + 0.748·4-s − 1.15·5-s − 2.02·6-s + 0.209·7-s + 0.332·8-s + 1.33·9-s + 1.52·10-s + 1.67·11-s + 1.14·12-s + 0.592·13-s − 0.276·14-s − 1.76·15-s − 1.18·16-s + 0.242·17-s − 1.76·18-s − 1.20·19-s − 0.864·20-s + 0.319·21-s − 2.21·22-s − 0.578·23-s + 0.508·24-s + 0.334·25-s − 0.783·26-s + 0.513·27-s + 0.156·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.206804561\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.206804561\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 - T \) |
| 47 | \( 1 + T \) |
good | 2 | \( 1 + 1.86T + 2T^{2} \) |
| 3 | \( 1 - 2.64T + 3T^{2} \) |
| 5 | \( 1 + 2.58T + 5T^{2} \) |
| 7 | \( 1 - 0.553T + 7T^{2} \) |
| 11 | \( 1 - 5.54T + 11T^{2} \) |
| 13 | \( 1 - 2.13T + 13T^{2} \) |
| 19 | \( 1 + 5.27T + 19T^{2} \) |
| 23 | \( 1 + 2.77T + 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 + 3.51T + 31T^{2} \) |
| 37 | \( 1 - 4.48T + 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 - 8.50T + 43T^{2} \) |
| 53 | \( 1 - 3.94T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 7.90T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 4.92T + 73T^{2} \) |
| 79 | \( 1 + 3.69T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 8.97T + 89T^{2} \) |
| 97 | \( 1 + 9.46T + 97T^{2} \) |
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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
−9.880991818885855235609765478119, −9.141000129098511547605476083834, −8.540918728292457865631662310643, −8.095570768808956659108102022616, −7.34140948060597375263484662445, −6.41810294225424577500962720921, −4.17848903421708572597040662827, −3.92058387976827729427153007095, −2.39448404870629072807111914734, −1.10534332824805437005439245374,
1.10534332824805437005439245374, 2.39448404870629072807111914734, 3.92058387976827729427153007095, 4.17848903421708572597040662827, 6.41810294225424577500962720921, 7.34140948060597375263484662445, 8.095570768808956659108102022616, 8.540918728292457865631662310643, 9.141000129098511547605476083834, 9.880991818885855235609765478119