| L(s) = 1 | + (−1.37 − 0.340i)2-s + (0.0980 − 0.995i)3-s + (1.76 + 0.935i)4-s + (2.97 − 1.59i)5-s + (−0.473 + 1.33i)6-s + (−0.855 − 4.30i)7-s + (−2.10 − 1.88i)8-s + (−0.980 − 0.195i)9-s + (−4.62 + 1.16i)10-s + (−2.53 + 3.09i)11-s + (1.10 − 1.66i)12-s + (2.25 − 4.22i)13-s + (−0.292 + 6.19i)14-s + (−1.29 − 3.11i)15-s + (2.24 + 3.30i)16-s + (0.483 − 1.16i)17-s + ⋯ |
| L(s) = 1 | + (−0.970 − 0.241i)2-s + (0.0565 − 0.574i)3-s + (0.883 + 0.467i)4-s + (1.33 − 0.711i)5-s + (−0.193 + 0.543i)6-s + (−0.323 − 1.62i)7-s + (−0.744 − 0.667i)8-s + (−0.326 − 0.0650i)9-s + (−1.46 + 0.369i)10-s + (−0.765 + 0.933i)11-s + (0.318 − 0.481i)12-s + (0.626 − 1.17i)13-s + (−0.0781 + 1.65i)14-s + (−0.333 − 0.805i)15-s + (0.562 + 0.827i)16-s + (0.117 − 0.283i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.459 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.525345 - 0.863144i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.525345 - 0.863144i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.37 + 0.340i)T \) |
| 3 | \( 1 + (-0.0980 + 0.995i)T \) |
| good | 5 | \( 1 + (-2.97 + 1.59i)T + (2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (0.855 + 4.30i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (2.53 - 3.09i)T + (-2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-2.25 + 4.22i)T + (-7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-0.483 + 1.16i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-1.78 - 5.88i)T + (-15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (6.40 - 4.28i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-0.780 + 0.640i)T + (5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (-4.72 + 4.72i)T - 31iT^{2} \) |
| 37 | \( 1 + (4.90 + 1.48i)T + (30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (-1.78 - 2.67i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (0.524 + 5.32i)T + (-42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (-5.85 - 2.42i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (2.39 + 1.96i)T + (10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (3.18 + 5.96i)T + (-32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (-13.2 - 1.30i)T + (59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-2.13 - 0.210i)T + (65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (-0.601 + 0.119i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (0.184 - 0.928i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-11.2 + 4.65i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-9.92 + 3.01i)T + (69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (-6.17 - 4.12i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (4.74 - 4.74i)T - 97iT^{2} \) |
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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
−10.53749788493754233272864054254, −10.11379674908172454789840481731, −9.499728575652537093994177960257, −8.033450099875563386500604294855, −7.64147631859402153535026660114, −6.42656491188234187471566860423, −5.48711865702449554725816901192, −3.67102148444882910959602233289, −2.06219654688179581877373640195, −0.921513309061948908616270784070,
2.18838822649388463947762501076, 2.91990651731598569745321537994, 5.28467541286877839842020270816, 6.07493371788445833954324509244, 6.64471845650324583451984753222, 8.402635022937634680804386721782, 8.948320407468674065460184731851, 9.700419312254967125409003980560, 10.50612985413496975462264773008, 11.27443344248699064421619253783
