L(s) = 1 | + (1.85 − 1.85i)2-s + (0.163 + 0.163i)3-s − 2.90i·4-s + (−4.36 + 2.43i)5-s + 0.608·6-s + (−1.87 + 1.87i)7-s + (2.02 + 2.02i)8-s − 8.94i·9-s + (−3.58 + 12.6i)10-s − 1.59·11-s + (0.476 − 0.476i)12-s + (2.82 + 2.82i)13-s + 6.95i·14-s + (−1.11 − 0.315i)15-s + 19.1·16-s + (−11.1 + 11.1i)17-s + ⋯ |
L(s) = 1 | + (0.929 − 0.929i)2-s + (0.0546 + 0.0546i)3-s − 0.727i·4-s + (−0.873 + 0.487i)5-s + 0.101·6-s + (−0.267 + 0.267i)7-s + (0.253 + 0.253i)8-s − 0.994i·9-s + (−0.358 + 1.26i)10-s − 0.145·11-s + (0.0397 − 0.0397i)12-s + (0.217 + 0.217i)13-s + 0.496i·14-s + (−0.0742 − 0.0210i)15-s + 1.19·16-s + (−0.656 + 0.656i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.29158 - 0.568919i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29158 - 0.568919i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (4.36 - 2.43i)T \) |
| 7 | \( 1 + (1.87 - 1.87i)T \) |
good | 2 | \( 1 + (-1.85 + 1.85i)T - 4iT^{2} \) |
| 3 | \( 1 + (-0.163 - 0.163i)T + 9iT^{2} \) |
| 11 | \( 1 + 1.59T + 121T^{2} \) |
| 13 | \( 1 + (-2.82 - 2.82i)T + 169iT^{2} \) |
| 17 | \( 1 + (11.1 - 11.1i)T - 289iT^{2} \) |
| 19 | \( 1 + 21.3iT - 361T^{2} \) |
| 23 | \( 1 + (13.8 + 13.8i)T + 529iT^{2} \) |
| 29 | \( 1 - 15.9iT - 841T^{2} \) |
| 31 | \( 1 - 53.2T + 961T^{2} \) |
| 37 | \( 1 + (45.5 - 45.5i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 65.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-48.4 - 48.4i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-10.9 + 10.9i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-48.0 - 48.0i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 53.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 22.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-4.35 + 4.35i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 38.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (20.2 + 20.2i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 90.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (74.1 + 74.1i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 108. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-48.6 + 48.6i)T - 9.40e3iT^{2} \) |
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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
−15.73161639832904242565358781143, −14.87569608088726815315928432422, −13.58948198054047732534052007645, −12.34183936713912623068152843809, −11.59506193350479455970615073837, −10.39509192955935506682776845047, −8.516576773338927315097192717928, −6.59367473078566157042915168819, −4.39605003776870971783246722804, −3.04129271191453024624644169006,
4.05554113768480104668837002815, 5.42335129651082195441798379869, 7.16589926513167747324724864784, 8.245883038019311314802486021420, 10.34120070709057114428600683168, 11.98581292654887861174670493253, 13.26412945812633242202267823133, 14.04738849722409845090969585343, 15.55533837647218106127150806079, 16.03225181802603325411664644396