| L(s) = 1 | + (1.33 + 0.473i)2-s + (0.248 − 0.124i)3-s + (1.55 + 1.26i)4-s + (−0.228 − 0.515i)5-s + (0.389 − 0.0489i)6-s + (3.03 + 1.81i)7-s + (1.47 + 2.41i)8-s + (−1.74 + 2.34i)9-s + (−0.0605 − 0.795i)10-s + (−2.20 − 1.72i)11-s + (0.542 + 0.119i)12-s + (1.36 − 1.29i)13-s + (3.18 + 3.86i)14-s + (−0.121 − 0.0993i)15-s + (0.816 + 3.91i)16-s + (0.453 + 0.552i)17-s + ⋯ |
| L(s) = 1 | + (0.942 + 0.334i)2-s + (0.143 − 0.0720i)3-s + (0.775 + 0.630i)4-s + (−0.102 − 0.230i)5-s + (0.159 − 0.0199i)6-s + (1.14 + 0.687i)7-s + (0.520 + 0.854i)8-s + (−0.580 + 0.782i)9-s + (−0.0191 − 0.251i)10-s + (−0.664 − 0.518i)11-s + (0.156 + 0.0343i)12-s + (0.378 − 0.360i)13-s + (0.851 + 1.03i)14-s + (−0.0312 − 0.0256i)15-s + (0.204 + 0.978i)16-s + (0.109 + 0.133i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.48922 + 1.11635i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.48922 + 1.11635i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.33 - 0.473i)T \) |
| good | 3 | \( 1 + (-0.248 + 0.124i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (0.228 + 0.515i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (-3.03 - 1.81i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (2.20 + 1.72i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (-1.36 + 1.29i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.453 - 0.552i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (0.309 + 1.78i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (-1.38 - 0.654i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (6.06 + 3.43i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (-0.394 + 0.0784i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (-5.10 - 3.23i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (-2.85 - 3.14i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (3.22 + 9.74i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (4.94 + 2.64i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (-0.696 + 0.394i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (7.75 - 8.14i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (0.933 + 12.6i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (-1.84 - 1.59i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (13.3 + 1.97i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (-8.57 + 5.14i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (16.5 + 5.00i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (-0.720 + 0.456i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (2.28 - 1.08i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (-13.2 - 8.84i)T + (37.1 + 89.6i)T^{2} \) |
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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
−11.25571396887174640840023705358, −10.53472997649517365600681337646, −8.782250465093544238433727666790, −8.183085141437559830609924302438, −7.54953143227495447019592097561, −6.07447288800820656769321890257, −5.31450620395107343790930441655, −4.61482123087327163667677155300, −3.09961757327985626629259033592, −2.05747215210107369089827811350,
1.46510920387648684157638958981, 2.93342495877437095815682985526, 4.02536553119837646972551153375, 4.93006429290617642749047340503, 5.95192662171172004576588323561, 7.08860761879657473376359645413, 7.84463576350708787027147164055, 9.116936597793192370099394635522, 10.17774039326260426861663063350, 11.14757332883834475563460294841
