| L(s) = 1 | + (−0.812 − 0.172i)2-s + (0.273 + 0.473i)3-s + (−1.19 − 0.532i)4-s + (−0.0415 − 0.395i)5-s + (−0.140 − 0.431i)6-s + (−2.05 + 1.66i)7-s + (2.22 + 1.61i)8-s + (1.35 − 2.33i)9-s + (−0.0345 + 0.328i)10-s + (0.343 − 3.26i)11-s + (−0.0748 − 0.712i)12-s + (−1.71 − 5.26i)13-s + (1.95 − 0.994i)14-s + (0.175 − 0.127i)15-s + (0.225 + 0.249i)16-s + (−0.0430 + 0.409i)17-s + ⋯ |
| L(s) = 1 | + (−0.574 − 0.122i)2-s + (0.157 + 0.273i)3-s + (−0.598 − 0.266i)4-s + (−0.0185 − 0.176i)5-s + (−0.0572 − 0.176i)6-s + (−0.778 + 0.628i)7-s + (0.786 + 0.571i)8-s + (0.450 − 0.779i)9-s + (−0.0109 + 0.103i)10-s + (0.103 − 0.985i)11-s + (−0.0216 − 0.205i)12-s + (−0.474 − 1.46i)13-s + (0.523 − 0.265i)14-s + (0.0454 − 0.0329i)15-s + (0.0562 + 0.0624i)16-s + (−0.0104 + 0.0993i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.368397 - 0.474820i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.368397 - 0.474820i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.05 - 1.66i)T \) |
| 41 | \( 1 + (-5.96 + 2.33i)T \) |
| good | 2 | \( 1 + (0.812 + 0.172i)T + (1.82 + 0.813i)T^{2} \) |
| 3 | \( 1 + (-0.273 - 0.473i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.0415 + 0.395i)T + (-4.89 + 1.03i)T^{2} \) |
| 11 | \( 1 + (-0.343 + 3.26i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (1.71 + 5.26i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.0430 - 0.409i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (3.39 + 3.77i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (7.12 + 1.51i)T + (21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (-0.893 + 0.649i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.527 + 5.01i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.953 - 9.07i)T + (-36.1 + 7.69i)T^{2} \) |
| 43 | \( 1 + (-0.919 - 2.83i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (8.60 + 1.82i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (5.63 + 2.50i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (1.53 - 1.70i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-9.49 - 10.5i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-11.3 - 5.06i)T + (44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (-6.41 - 4.66i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.361 - 0.625i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.28 - 2.22i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + (5.80 + 6.45i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-1.57 + 1.14i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.39106776348613179385022258710, −10.18847482052453323923999972967, −9.759897993227777459379633022302, −8.717668588588827422649939731997, −8.161811322028822909956169555732, −6.51120509041501020622796631159, −5.52411316831952964122564638388, −4.22127148659405196997248233476, −2.84898624364015514261609612727, −0.54806049854431952356226295340,
1.89647370167275392989027784431, 3.89001914390204911683163945929, 4.70252556019489494075728042293, 6.62718575026106941769343198268, 7.30732615521407705260009181480, 8.175215602407205039315971108861, 9.426660305240732536271811680864, 9.931173764090296024680718640567, 10.86836265754852007249628308063, 12.46839299864859828053246542162
