L(s) = 1 | + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + (0.5 + 2.59i)7-s + (−0.499 − 0.866i)9-s + (−1.5 + 2.59i)11-s + 4·13-s − 0.999·15-s + (2 + 3.46i)19-s + (−2.5 − 0.866i)21-s + (−4 − 6.92i)23-s + (2 − 3.46i)25-s + 0.999·27-s − 3·29-s + (2.5 − 4.33i)31-s + (−1.5 − 2.59i)33-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (0.223 + 0.387i)5-s + (0.188 + 0.981i)7-s + (−0.166 − 0.288i)9-s + (−0.452 + 0.783i)11-s + 1.10·13-s − 0.258·15-s + (0.458 + 0.794i)19-s + (−0.545 − 0.188i)21-s + (−0.834 − 1.44i)23-s + (0.400 − 0.692i)25-s + 0.192·27-s − 0.557·29-s + (0.449 − 0.777i)31-s + (−0.261 − 0.452i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.902111 + 0.600068i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.902111 + 0.600068i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.5 - 2.59i)T \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + (5 + 8.66i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.5 + 4.33i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3 - 5.19i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7T + 83T^{2} \) |
| 89 | \( 1 + (-9 - 15.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 17T + 97T^{2} \) |
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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
−12.77700785914237666091944282905, −11.99571640242203150594276245619, −10.89161745293403720966204279901, −10.08768790591763689599556331058, −9.000423049924756827261221646976, −7.942831106571960374212948670689, −6.38018132140269793665373088371, −5.51090442417589854718756050162, −4.10350619475580917350151538029, −2.39487519051647458122973577984,
1.23009030174167561306806809656, 3.46525377428093816798725622350, 5.06532804769675224381579077430, 6.19750591618151433494307400364, 7.42004411163894806630952604738, 8.362824526134932600180029505650, 9.590736515814706101220057799992, 10.90354637271078615565653728121, 11.41031791247393288423130568905, 12.83942181260212742152509582866