| L(s) = 1 | + (−0.809 + 0.587i)2-s + (−0.351 + 1.08i)3-s + (0.309 − 0.951i)4-s + (−0.646 + 2.14i)5-s + (−0.351 − 1.08i)6-s − 1.02·7-s + (0.309 + 0.951i)8-s + (1.38 + 1.00i)9-s + (−0.734 − 2.11i)10-s + (−3.22 + 2.34i)11-s + (0.919 + 0.667i)12-s + (−4.54 − 3.30i)13-s + (0.825 − 0.599i)14-s + (−2.08 − 1.45i)15-s + (−0.809 − 0.587i)16-s + (−0.120 − 0.370i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.202 + 0.623i)3-s + (0.154 − 0.475i)4-s + (−0.289 + 0.957i)5-s + (−0.143 − 0.441i)6-s − 0.385·7-s + (0.109 + 0.336i)8-s + (0.460 + 0.334i)9-s + (−0.232 − 0.667i)10-s + (−0.971 + 0.705i)11-s + (0.265 + 0.192i)12-s + (−1.26 − 0.916i)13-s + (0.220 − 0.160i)14-s + (−0.538 − 0.374i)15-s + (−0.202 − 0.146i)16-s + (−0.0291 − 0.0897i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0622300 - 0.0721202i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0622300 - 0.0721202i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (0.646 - 2.14i)T \) |
| 19 | \( 1 + (0.309 + 0.951i)T \) |
| good | 3 | \( 1 + (0.351 - 1.08i)T + (-2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + 1.02T + 7T^{2} \) |
| 11 | \( 1 + (3.22 - 2.34i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (4.54 + 3.30i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.120 + 0.370i)T + (-13.7 + 9.99i)T^{2} \) |
| 23 | \( 1 + (-5.62 + 4.08i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (1.31 - 4.06i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.16 - 6.66i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (4.28 + 3.11i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (2.21 + 1.61i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 0.0803T + 43T^{2} \) |
| 47 | \( 1 + (-3.74 + 11.5i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.983 - 3.02i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-2.90 - 2.10i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.499 + 0.363i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (4.88 + 15.0i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-4.12 + 12.6i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (1.37 - 0.997i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.476 + 1.46i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.75 - 8.46i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (10.5 - 7.69i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.51 + 13.9i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.44959628071566904731476796107, −10.06114820710734774166250295257, −9.119882122269757353118521851632, −7.955721968623632172303088414729, −7.23492261562355044893303299825, −6.75092454364272962980054540527, −5.28494588055986330207284507300, −4.80656532889286311128482495491, −3.29791453799138460123279072122, −2.30196950280790499601208368140,
0.05623779011183666747212570592, 1.33724730114245096469769761826, 2.61857829176235391649897025046, 3.95031526126890733571928977574, 4.97512593296250971866038752812, 6.07605897079200029771111010988, 7.17612632767503754374737301263, 7.74009618083003805813558453902, 8.640135646465186998236246238482, 9.553090244440127387574880889338
