| L(s) = 1 | + (−0.775 + 1.52i)2-s + (−0.270 + 1.71i)3-s + (0.634 + 0.873i)4-s + (4.50 + 2.17i)5-s + (−2.39 − 1.73i)6-s + (−3.94 − 3.94i)7-s + (−8.57 + 1.35i)8-s + (−2.85 − 0.927i)9-s + (−6.80 + 5.16i)10-s + (6.15 + 18.9i)11-s + (−1.66 + 0.849i)12-s + (−3.61 − 7.10i)13-s + (9.07 − 2.94i)14-s + (−4.94 + 7.11i)15-s + (3.24 − 9.99i)16-s + (−1.96 − 12.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.387 + 0.761i)2-s + (−0.0903 + 0.570i)3-s + (0.158 + 0.218i)4-s + (0.900 + 0.435i)5-s + (−0.399 − 0.289i)6-s + (−0.563 − 0.563i)7-s + (−1.07 + 0.169i)8-s + (−0.317 − 0.103i)9-s + (−0.680 + 0.516i)10-s + (0.559 + 1.72i)11-s + (−0.138 + 0.0707i)12-s + (−0.278 − 0.546i)13-s + (0.648 − 0.210i)14-s + (−0.329 + 0.474i)15-s + (0.203 − 0.624i)16-s + (−0.115 − 0.727i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.544 - 0.838i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.544 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.532132 + 0.980068i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.532132 + 0.980068i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.270 - 1.71i)T \) |
| 5 | \( 1 + (-4.50 - 2.17i)T \) |
| good | 2 | \( 1 + (0.775 - 1.52i)T + (-2.35 - 3.23i)T^{2} \) |
| 7 | \( 1 + (3.94 + 3.94i)T + 49iT^{2} \) |
| 11 | \( 1 + (-6.15 - 18.9i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (3.61 + 7.10i)T + (-99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (1.96 + 12.3i)T + (-274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (-15.6 + 21.5i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-36.0 - 18.3i)T + (310. + 427. i)T^{2} \) |
| 29 | \( 1 + (11.8 + 16.3i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-25.8 - 18.7i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-5.53 + 2.81i)T + (804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-8.80 + 27.0i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (24.5 - 24.5i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-5.29 - 0.838i)T + (2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (1.08 - 6.88i)T + (-2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (91.6 + 29.7i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (21.1 + 65.0i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (8.49 + 53.6i)T + (-4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (10.0 - 7.30i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-58.6 - 29.9i)T + (3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-42.5 - 58.5i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (101. - 16.0i)T + (6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (49.5 - 16.1i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (66.1 + 10.4i)T + (8.94e3 + 2.90e3i)T^{2} \) |
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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
−15.06806989466922538300301300443, −13.78010586696980610288889283012, −12.56084718917874449943583977850, −11.20717568823576435587265326514, −9.754214834675476720949389438935, −9.318456225447654209171131286655, −7.33041935444044876049759161989, −6.70473862068309495146751087358, −5.07105445469566663366092670380, −2.97836757368331470074013222354,
1.25439147538414168820408334432, 2.93621638586062602176470482488, 5.73607938972473264014532202156, 6.42460035606850012300026303386, 8.632555596680225121164934987240, 9.382793774159601346325178614318, 10.63002397795237619380368761603, 11.72596844309499762023897828603, 12.64451424107008543757817825190, 13.72725407423793105591786676461
