L(s) = 1 | − 2.84·5-s − 3.53·7-s − 4.06·11-s − 4.24i·13-s + 5.67i·17-s + 0.496i·19-s + 3.35·23-s + 3.10·25-s + 3.85·29-s − 0.715i·31-s + 10.0·35-s + 10.1·37-s − 0.201·41-s − 9.49i·43-s + (−3.15 + 6.08i)47-s + ⋯ |
L(s) = 1 | − 1.27·5-s − 1.33·7-s − 1.22·11-s − 1.17i·13-s + 1.37i·17-s + 0.113i·19-s + 0.700·23-s + 0.621·25-s + 0.716·29-s − 0.128i·31-s + 1.70·35-s + 1.66·37-s − 0.0315·41-s − 1.44i·43-s + (−0.460 + 0.887i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1692 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1692 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7195226528\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7195226528\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 47 | \( 1 + (3.15 - 6.08i)T \) |
good | 5 | \( 1 + 2.84T + 5T^{2} \) |
| 7 | \( 1 + 3.53T + 7T^{2} \) |
| 11 | \( 1 + 4.06T + 11T^{2} \) |
| 13 | \( 1 + 4.24iT - 13T^{2} \) |
| 17 | \( 1 - 5.67iT - 17T^{2} \) |
| 19 | \( 1 - 0.496iT - 19T^{2} \) |
| 23 | \( 1 - 3.35T + 23T^{2} \) |
| 29 | \( 1 - 3.85T + 29T^{2} \) |
| 31 | \( 1 + 0.715iT - 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 0.201T + 41T^{2} \) |
| 43 | \( 1 + 9.49iT - 43T^{2} \) |
| 53 | \( 1 - 3.66iT - 53T^{2} \) |
| 59 | \( 1 + 9.26iT - 59T^{2} \) |
| 61 | \( 1 - 8.60T + 61T^{2} \) |
| 67 | \( 1 - 13.8iT - 67T^{2} \) |
| 71 | \( 1 - 4.39iT - 71T^{2} \) |
| 73 | \( 1 + 7.98iT - 73T^{2} \) |
| 79 | \( 1 - 5.53T + 79T^{2} \) |
| 83 | \( 1 + 2.85iT - 83T^{2} \) |
| 89 | \( 1 - 7.77iT - 89T^{2} \) |
| 97 | \( 1 + 4.96T + 97T^{2} \) |
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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
−9.397011960172762334379260072651, −8.248670668574440131243122033828, −7.974410176777568417323203761512, −7.05878865382162661581496368103, −6.15377806308008670619016734035, −5.32852127734006151118489228873, −4.17814283112447701339191146672, −3.37715255467352710987925375862, −2.66409202908218964719721841817, −0.59140789104505726730238876397,
0.52467597757697256279630802032, 2.60520127249378066443753599169, 3.27088279798742499622900846440, 4.32000790320876233315058785862, 5.05594669482318339534588281730, 6.30529857592330041730086808562, 7.02089902978941212302469609383, 7.64601322660076155909881318913, 8.497958105283252646991820372753, 9.420702068098427412839583861255