| L(s) = 1 | + (−116. − 201. i)2-s + (2.61e3 + 1.50e3i)3-s + (5.70e3 − 9.88e3i)4-s + (−3.01e5 + 1.73e5i)5-s − 7.01e5i·6-s + (2.93e6 − 4.96e6i)7-s − 1.79e7·8-s + (−1.69e7 − 2.94e7i)9-s + (7.01e7 + 4.04e7i)10-s + (−1.55e8 + 2.69e8i)11-s + (2.97e7 − 1.72e7i)12-s + 1.33e9i·13-s + (−1.34e9 − 1.34e7i)14-s − 1.04e9·15-s + (1.70e9 + 2.95e9i)16-s + (−6.18e9 − 3.57e9i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 − 0.787i)2-s + (0.397 + 0.229i)3-s + (0.0870 − 0.150i)4-s + (−0.771 + 0.445i)5-s − 0.417i·6-s + (0.508 − 0.860i)7-s − 1.06·8-s + (−0.394 − 0.683i)9-s + (0.701 + 0.404i)10-s + (−0.726 + 1.25i)11-s + (0.0692 − 0.0400i)12-s + 1.63i·13-s + (−0.908 − 0.00909i)14-s − 0.409·15-s + (0.397 + 0.688i)16-s + (−0.886 − 0.512i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 - 0.801i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.597 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{17}{2})\) |
\(\approx\) |
\(0.0196809 + 0.0392069i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0196809 + 0.0392069i\) |
| \(L(9)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.93e6 + 4.96e6i)T \) |
| good | 2 | \( 1 + (116. + 201. i)T + (-3.27e4 + 5.67e4i)T^{2} \) |
| 3 | \( 1 + (-2.61e3 - 1.50e3i)T + (2.15e7 + 3.72e7i)T^{2} \) |
| 5 | \( 1 + (3.01e5 - 1.73e5i)T + (7.62e10 - 1.32e11i)T^{2} \) |
| 11 | \( 1 + (1.55e8 - 2.69e8i)T + (-2.29e16 - 3.97e16i)T^{2} \) |
| 13 | \( 1 - 1.33e9iT - 6.65e17T^{2} \) |
| 17 | \( 1 + (6.18e9 + 3.57e9i)T + (2.43e19 + 4.21e19i)T^{2} \) |
| 19 | \( 1 + (4.41e9 - 2.54e9i)T + (1.44e20 - 2.49e20i)T^{2} \) |
| 23 | \( 1 + (1.95e10 + 3.38e10i)T + (-3.06e21 + 5.31e21i)T^{2} \) |
| 29 | \( 1 + 5.16e11T + 2.50e23T^{2} \) |
| 31 | \( 1 + (-4.84e11 - 2.79e11i)T + (3.63e23 + 6.29e23i)T^{2} \) |
| 37 | \( 1 + (2.36e12 + 4.10e12i)T + (-6.16e24 + 1.06e25i)T^{2} \) |
| 41 | \( 1 + 1.21e13iT - 6.37e25T^{2} \) |
| 43 | \( 1 - 7.60e12T + 1.36e26T^{2} \) |
| 47 | \( 1 + (-1.03e13 + 5.98e12i)T + (2.83e26 - 4.91e26i)T^{2} \) |
| 53 | \( 1 + (4.19e13 - 7.26e13i)T + (-1.93e27 - 3.35e27i)T^{2} \) |
| 59 | \( 1 + (-9.75e13 - 5.63e13i)T + (1.07e28 + 1.86e28i)T^{2} \) |
| 61 | \( 1 + (-4.98e13 + 2.87e13i)T + (1.83e28 - 3.18e28i)T^{2} \) |
| 67 | \( 1 + (-1.75e14 + 3.04e14i)T + (-8.24e28 - 1.42e29i)T^{2} \) |
| 71 | \( 1 + 2.35e14T + 4.16e29T^{2} \) |
| 73 | \( 1 + (8.72e14 + 5.03e14i)T + (3.25e29 + 5.63e29i)T^{2} \) |
| 79 | \( 1 + (1.44e14 + 2.50e14i)T + (-1.15e30 + 1.99e30i)T^{2} \) |
| 83 | \( 1 + 5.23e14iT - 5.07e30T^{2} \) |
| 89 | \( 1 + (6.58e14 - 3.80e14i)T + (7.74e30 - 1.34e31i)T^{2} \) |
| 97 | \( 1 + 8.13e14iT - 6.14e31T^{2} \) |
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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
−17.67849219696674387702540001605, −15.50802324162691125596782373899, −14.34672897422471422634003242993, −11.92900682249273322708371347193, −10.73589751249090124172486459760, −9.192288857141180518990356286160, −7.09698577624453749239439273985, −4.06140990717624237873068089555, −2.11605503244850693734409938319, −0.02013027543756066870406925832,
2.84890674318890231489839155866, 5.61910339339107360814236854608, 8.091249146172483844464212226068, 8.353209362585147610900203995819, 11.31762560508790395812516745746, 13.02003471004327308483309638023, 15.14930088487902815006921454184, 16.06026834957050951904788063074, 17.60488887022583736668213717075, 19.01403656272801052431954533597
