| L(s) = 1 | + (−0.142 + 0.989i)2-s + (−0.415 − 0.909i)3-s + (−0.959 − 0.281i)4-s + (−0.654 − 0.755i)5-s + (0.959 − 0.281i)6-s + (−1.32 − 0.850i)7-s + (0.415 − 0.909i)8-s + (−0.654 + 0.755i)9-s + (0.841 − 0.540i)10-s + (0.308 + 2.14i)11-s + (0.142 + 0.989i)12-s + (−0.178 + 0.114i)13-s + (1.03 − 1.18i)14-s + (−0.415 + 0.909i)15-s + (0.841 + 0.540i)16-s + (−2.86 + 0.841i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (−0.239 − 0.525i)3-s + (−0.479 − 0.140i)4-s + (−0.292 − 0.337i)5-s + (0.391 − 0.115i)6-s + (−0.500 − 0.321i)7-s + (0.146 − 0.321i)8-s + (−0.218 + 0.251i)9-s + (0.266 − 0.170i)10-s + (0.0930 + 0.646i)11-s + (0.0410 + 0.285i)12-s + (−0.0494 + 0.0317i)13-s + (0.275 − 0.317i)14-s + (−0.107 + 0.234i)15-s + (0.210 + 0.135i)16-s + (−0.694 + 0.204i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0557770 + 0.310790i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0557770 + 0.310790i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 3 | \( 1 + (0.415 + 0.909i)T \) |
| 5 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (2.44 - 4.12i)T \) |
| good | 7 | \( 1 + (1.32 + 0.850i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.308 - 2.14i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (0.178 - 0.114i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (2.86 - 0.841i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-0.700 - 0.205i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (5.97 - 1.75i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (1.76 - 3.85i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (6.99 - 8.07i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-6.89 - 7.95i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (3.95 + 8.66i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 5.12T + 47T^{2} \) |
| 53 | \( 1 + (-6.72 - 4.32i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (6.90 - 4.43i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (2.36 - 5.18i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-2.17 + 15.1i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.286 + 1.99i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (1.77 + 0.521i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-2.50 + 1.61i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (3.99 - 4.60i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-5.73 - 12.5i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (5.45 + 6.29i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.78991235751116662262041340503, −9.827688657338472424959846194571, −9.025333244904153711691029239675, −8.083120274725067126131538455022, −7.24297246518772912193307025753, −6.63044850823062452629399888042, −5.58359528481247678021453182678, −4.61994895889375575989726265941, −3.47850525722306153886752552005, −1.64272998515101555861153488876,
0.17424118381430647485102155474, 2.31019545977913703573442419808, 3.43219833783916458971512229988, 4.25963323600729088937181980101, 5.49055242674043968279811002159, 6.38589869197087858910898179846, 7.56990829472738946970679310448, 8.681304153502256342054416270661, 9.334051236729582035296615719700, 10.18112248826408535464950866048
