| L(s) = 1 | + (0.786 − 0.618i)2-s + (1.31 − 0.933i)3-s + (0.235 − 0.971i)4-s + (1.08 + 0.208i)5-s + (0.453 − 1.54i)6-s + (0.523 + 2.59i)7-s + (−0.415 − 0.909i)8-s + (−0.133 + 0.385i)9-s + (0.981 − 0.505i)10-s + (1.95 − 2.48i)11-s + (−0.598 − 1.49i)12-s + (0.970 − 1.51i)13-s + (2.01 + 1.71i)14-s + (1.61 − 0.738i)15-s + (−0.888 − 0.458i)16-s + (−2.88 + 2.74i)17-s + ⋯ |
| L(s) = 1 | + (0.555 − 0.437i)2-s + (0.757 − 0.539i)3-s + (0.117 − 0.485i)4-s + (0.484 + 0.0934i)5-s + (0.185 − 0.630i)6-s + (0.197 + 0.980i)7-s + (−0.146 − 0.321i)8-s + (−0.0444 + 0.128i)9-s + (0.310 − 0.159i)10-s + (0.589 − 0.749i)11-s + (−0.172 − 0.431i)12-s + (0.269 − 0.418i)13-s + (0.538 + 0.458i)14-s + (0.417 − 0.190i)15-s + (−0.222 − 0.114i)16-s + (−0.699 + 0.666i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.08232 - 1.00178i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.08232 - 1.00178i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.786 + 0.618i)T \) |
| 7 | \( 1 + (-0.523 - 2.59i)T \) |
| 23 | \( 1 + (4.03 - 2.59i)T \) |
| good | 3 | \( 1 + (-1.31 + 0.933i)T + (0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (-1.08 - 0.208i)T + (4.64 + 1.85i)T^{2} \) |
| 11 | \( 1 + (-1.95 + 2.48i)T + (-2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-0.970 + 1.51i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (2.88 - 2.74i)T + (0.808 - 16.9i)T^{2} \) |
| 19 | \( 1 + (4.50 + 4.29i)T + (0.904 + 18.9i)T^{2} \) |
| 29 | \( 1 + (6.53 + 1.91i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.0568 + 0.595i)T + (-30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-8.52 - 2.94i)T + (29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (-0.356 - 0.309i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-9.92 - 4.53i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (0.473 - 0.273i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.38 + 0.399i)T + (52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (0.775 + 1.50i)T + (-34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (4.92 - 6.90i)T + (-19.9 - 57.6i)T^{2} \) |
| 67 | \( 1 + (-4.12 + 10.3i)T + (-48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (-0.259 - 1.80i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (11.2 + 2.73i)T + (64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-15.3 - 0.732i)T + (78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-0.215 - 0.248i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (11.0 - 1.05i)T + (87.3 - 16.8i)T^{2} \) |
| 97 | \( 1 + (6.03 - 6.96i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.47403215881630443250260270218, −10.85930218551759073772205914917, −9.481461664369090531728026751185, −8.720896897011029177746071021737, −7.84330441995776971690791623631, −6.31846562641141045215516112429, −5.68901145478325455939447052058, −4.15961714356405600970921504489, −2.72788360617852087865967107962, −1.89929986199725290414276718394,
2.15363017284987580856601912319, 3.92984744492528427919343906331, 4.27342251302685907087757553630, 5.89409514163455114137528249290, 6.88555261264482856141256649991, 7.904260200939748404118817460558, 9.015948650210424702475423001778, 9.713754156397337673084362727530, 10.75217302809252661082871778686, 11.87082434836532013542517337444
