L(s) = 1 | + (0.562 − 0.973i)2-s + (−0.0186 − 1.73i)3-s + (0.368 + 0.637i)4-s + (0.5 + 0.866i)5-s + (−1.69 − 0.955i)6-s + (1.91 − 3.31i)7-s + 3.07·8-s + (−2.99 + 0.0646i)9-s + 1.12·10-s + (−0.975 + 1.68i)11-s + (1.09 − 0.649i)12-s + (−0.5 − 0.866i)13-s + (−2.15 − 3.72i)14-s + (1.49 − 0.882i)15-s + (0.992 − 1.71i)16-s + 6.58·17-s + ⋯ |
L(s) = 1 | + (0.397 − 0.688i)2-s + (−0.0107 − 0.999i)3-s + (0.184 + 0.318i)4-s + (0.223 + 0.387i)5-s + (−0.692 − 0.389i)6-s + (0.723 − 1.25i)7-s + 1.08·8-s + (−0.999 + 0.0215i)9-s + 0.355·10-s + (−0.294 + 0.509i)11-s + (0.316 − 0.187i)12-s + (−0.138 − 0.240i)13-s + (−0.575 − 0.996i)14-s + (0.384 − 0.227i)15-s + (0.248 − 0.429i)16-s + 1.59·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.152 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.152 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.40970 - 1.64369i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.40970 - 1.64369i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.0186 + 1.73i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-0.562 + 0.973i)T + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (-1.91 + 3.31i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.975 - 1.68i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 6.58T + 17T^{2} \) |
| 19 | \( 1 - 0.304T + 19T^{2} \) |
| 23 | \( 1 + (3.00 + 5.21i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.378 - 0.654i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.95 + 6.85i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 8.54T + 37T^{2} \) |
| 41 | \( 1 + (-5.74 - 9.94i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.06 - 5.30i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.90 + 8.49i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 0.480T + 53T^{2} \) |
| 59 | \( 1 + (-4.91 - 8.50i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.93 - 8.55i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.805 - 1.39i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + (-3.35 + 5.81i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.46 - 7.72i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 6.53T + 89T^{2} \) |
| 97 | \( 1 + (-0.256 + 0.443i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.57305624705994180195533800276, −10.07467608380309359535462904416, −8.362988245137895936760180530432, −7.46837880845990171023589665608, −7.30014352070185455458258281560, −5.91204334912969278089268573191, −4.64686412859684278610194878714, −3.51014730298807307162176230708, −2.38566385779207632731782644222, −1.25262421664218385120880258821,
1.85838662342732895960432477160, 3.42430174597547042302990662507, 4.80674909672903283363906416107, 5.62580225111223276241319886021, 5.72496818776035274594331295535, 7.40316666348571508418545416691, 8.367468632438499517744308491872, 9.142082019949388258444389798694, 10.06332884067747106107153530092, 10.84877093756190182339247728177