L(s) = 1 | + (0.570 + 1.63i)3-s + (0.764 + 1.32i)5-s + (−1.91 + 1.82i)7-s + (−2.34 + 1.86i)9-s + (−0.417 + 0.723i)11-s + (1.81 − 3.13i)13-s + (−1.73 + 2.00i)15-s + (0.301 + 0.521i)17-s + (0.846 − 1.46i)19-s + (−4.07 − 2.10i)21-s + (3.07 + 5.32i)23-s + (1.33 − 2.30i)25-s + (−4.39 − 2.77i)27-s + (4.99 + 8.65i)29-s − 3.30·31-s + ⋯ |
L(s) = 1 | + (0.329 + 0.944i)3-s + (0.341 + 0.592i)5-s + (−0.725 + 0.688i)7-s + (−0.783 + 0.621i)9-s + (−0.125 + 0.218i)11-s + (0.502 − 0.870i)13-s + (−0.446 + 0.517i)15-s + (0.0730 + 0.126i)17-s + (0.194 − 0.336i)19-s + (−0.888 − 0.458i)21-s + (0.640 + 1.10i)23-s + (0.266 − 0.460i)25-s + (−0.845 − 0.534i)27-s + (0.927 + 1.60i)29-s − 0.594·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0969 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0969 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.880966 + 0.970916i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.880966 + 0.970916i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.570 - 1.63i)T \) |
| 7 | \( 1 + (1.91 - 1.82i)T \) |
good | 5 | \( 1 + (-0.764 - 1.32i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.417 - 0.723i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.81 + 3.13i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.301 - 0.521i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.846 + 1.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.07 - 5.32i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.99 - 8.65i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 3.30T + 31T^{2} \) |
| 37 | \( 1 + (-4.39 + 7.61i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.51 + 6.09i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.846 - 1.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 8.46T + 47T^{2} \) |
| 53 | \( 1 + (3.99 + 6.92i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 0.130T + 59T^{2} \) |
| 61 | \( 1 + 4.76T + 61T^{2} \) |
| 67 | \( 1 + 2.24T + 67T^{2} \) |
| 71 | \( 1 + 9.39T + 71T^{2} \) |
| 73 | \( 1 + (2.25 + 3.90i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 + (3.16 + 5.47i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.531 - 0.920i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.76 + 13.4i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.33543573239288735432158544036, −10.99101910132092855427080880747, −10.42479256904794417937465008827, −9.408031047378261620488675019008, −8.737403478007060848085126398025, −7.39282946554522678841609285998, −6.04913899756558406661469428812, −5.16853409888378652985369723602, −3.54185640376500686899125784813, −2.65448382001076245664708238447,
1.10161820228680079339764307731, 2.83420961613768087635905577744, 4.32903839748405230085888996345, 5.97574289874586862929175447000, 6.74518203923294113487237008300, 7.84390508959387679713340537353, 8.864984695036209799484540268164, 9.666351726428150926606385573158, 10.93110500971199653888708572371, 12.01314329141057827876269646652