| L(s) = 1 | − 1.77i·3-s + 1.91i·5-s + 2.45i·7-s − 0.167·9-s − 1.63i·11-s + 2.53·13-s + 3.40·15-s + (−3.23 + 2.56i)17-s + 2.17·19-s + 4.37·21-s − 2.41i·23-s + 1.33·25-s − 5.04i·27-s + 4.33i·29-s + 5.11i·31-s + ⋯ |
| L(s) = 1 | − 1.02i·3-s + 0.856i·5-s + 0.928i·7-s − 0.0558·9-s − 0.491i·11-s + 0.704·13-s + 0.880·15-s + (−0.783 + 0.621i)17-s + 0.498·19-s + 0.954·21-s − 0.503i·23-s + 0.266·25-s − 0.970i·27-s + 0.804i·29-s + 0.918i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.864460932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.864460932\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (3.23 - 2.56i)T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 1.77iT - 3T^{2} \) |
| 5 | \( 1 - 1.91iT - 5T^{2} \) |
| 7 | \( 1 - 2.45iT - 7T^{2} \) |
| 11 | \( 1 + 1.63iT - 11T^{2} \) |
| 13 | \( 1 - 2.53T + 13T^{2} \) |
| 19 | \( 1 - 2.17T + 19T^{2} \) |
| 23 | \( 1 + 2.41iT - 23T^{2} \) |
| 29 | \( 1 - 4.33iT - 29T^{2} \) |
| 31 | \( 1 - 5.11iT - 31T^{2} \) |
| 37 | \( 1 - 4.50iT - 37T^{2} \) |
| 41 | \( 1 + 4.47iT - 41T^{2} \) |
| 43 | \( 1 + 4.13T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 14.4T + 53T^{2} \) |
| 61 | \( 1 + 3.02iT - 61T^{2} \) |
| 67 | \( 1 - 6.34T + 67T^{2} \) |
| 71 | \( 1 - 6.97iT - 71T^{2} \) |
| 73 | \( 1 - 9.10iT - 73T^{2} \) |
| 79 | \( 1 + 4.05iT - 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 - 5.92T + 89T^{2} \) |
| 97 | \( 1 - 11.7iT - 97T^{2} \) |
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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
−8.612840090305858080486390623341, −7.71050569726112489459647809525, −6.91122972039725955187434825641, −6.46431610412523386281216935462, −5.84650869915579051140917220073, −4.89703228214441886119305250658, −3.71073304836243597172930763992, −2.87689905225106413859637170249, −2.09738271348734063609270688230, −1.11013265196231574592456698743,
0.60849749920369696299247274356, 1.71947121078350078865393426569, 3.10760588909204238712164128215, 4.09372519877221438687307638493, 4.41843415401709568711107272889, 5.13716651306453738123053359581, 6.01675937811554722999638719370, 7.03798312202560320906851318073, 7.62463611544469960331722423021, 8.514972628153000128612404101806
