| L(s) = 1 | + (0.620 + 0.450i)2-s + (−0.104 − 0.994i)3-s + (−0.436 − 1.34i)4-s + (0.582 + 1.00i)5-s + (0.383 − 0.664i)6-s + (3.17 + 0.675i)7-s + (0.808 − 2.48i)8-s + (−0.978 + 0.207i)9-s + (−0.0934 + 0.888i)10-s + (−3.57 + 3.96i)11-s + (−1.28 + 0.574i)12-s + (−5.67 − 2.52i)13-s + (1.66 + 1.85i)14-s + (0.942 − 0.685i)15-s + (−0.660 + 0.480i)16-s + (3.76 + 4.18i)17-s + ⋯ |
| L(s) = 1 | + (0.438 + 0.318i)2-s + (−0.0603 − 0.574i)3-s + (−0.218 − 0.671i)4-s + (0.260 + 0.451i)5-s + (0.156 − 0.271i)6-s + (1.20 + 0.255i)7-s + (0.285 − 0.879i)8-s + (−0.326 + 0.0693i)9-s + (−0.0295 + 0.281i)10-s + (−1.07 + 1.19i)11-s + (−0.372 + 0.165i)12-s + (−1.57 − 0.700i)13-s + (0.445 + 0.494i)14-s + (0.243 − 0.176i)15-s + (−0.165 + 0.120i)16-s + (0.912 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.18622 - 0.157774i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18622 - 0.157774i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.104 + 0.994i)T \) |
| 31 | \( 1 + (-4.32 - 3.51i)T \) |
| good | 2 | \( 1 + (-0.620 - 0.450i)T + (0.618 + 1.90i)T^{2} \) |
| 5 | \( 1 + (-0.582 - 1.00i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-3.17 - 0.675i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (3.57 - 3.96i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (5.67 + 2.52i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-3.76 - 4.18i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (1.40 - 0.623i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (0.465 - 1.43i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (1.33 + 0.967i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (-2.15 + 3.72i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.06 + 10.1i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (2.21 - 0.987i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (0.950 - 0.690i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (3.10 - 0.660i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (0.503 + 4.79i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 + (0.754 + 1.30i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (7.38 - 1.56i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (0.105 - 0.116i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (0.173 + 0.193i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (0.510 - 4.85i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (4.85 + 14.9i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-3.95 - 12.1i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−14.39971528129520287115179173021, −12.93995666570999874890220397578, −12.21452095405280328708618566941, −10.58672744214180428694900389522, −9.949373327229068771825695715805, −8.070799267659497751824535628900, −7.15308165171385187933914888432, −5.63509443237454329439997243339, −4.79016211486700681814033406913, −2.13721756044969943610684277999,
2.79828941489530674079306359362, 4.61752927818085026421150239364, 5.23846489562115919469385129440, 7.60579090730915646850545507460, 8.484023044134712837562383117608, 9.796883562810417528301835815981, 11.18204793578509311437885300894, 11.83694552439164434541968063385, 13.10049536605089518528119659808, 14.02771621332692537582474998373
