| L(s) = 1 | + (1.82 + 0.209i)2-s + (2.66 + 1.93i)3-s + (1.34 + 0.311i)4-s + (−0.564 + 0.825i)5-s + (4.46 + 4.09i)6-s + (0.0611 + 0.0629i)7-s + (−1.07 − 0.385i)8-s + (2.43 + 7.49i)9-s + (−1.20 + 1.38i)10-s + (−1.66 − 2.86i)11-s + (2.97 + 3.43i)12-s + (2.98 − 4.94i)13-s + (0.0984 + 0.127i)14-s + (−3.10 + 1.10i)15-s + (−4.35 − 2.14i)16-s + (2.50 − 2.04i)17-s + ⋯ |
| L(s) = 1 | + (1.29 + 0.148i)2-s + (1.54 + 1.11i)3-s + (0.670 + 0.155i)4-s + (−0.252 + 0.369i)5-s + (1.82 + 1.67i)6-s + (0.0231 + 0.0237i)7-s + (−0.381 − 0.136i)8-s + (0.811 + 2.49i)9-s + (−0.380 + 0.439i)10-s + (−0.502 − 0.864i)11-s + (0.858 + 0.990i)12-s + (0.826 − 1.37i)13-s + (0.0263 + 0.0341i)14-s + (−0.801 + 0.286i)15-s + (−1.08 − 0.535i)16-s + (0.607 − 0.497i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.35653 + 2.25691i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.35653 + 2.25691i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.564 - 0.825i)T \) |
| 11 | \( 1 + (1.66 + 2.86i)T \) |
| good | 2 | \( 1 + (-1.82 - 0.209i)T + (1.94 + 0.452i)T^{2} \) |
| 3 | \( 1 + (-2.66 - 1.93i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + (-0.0611 - 0.0629i)T + (-0.199 + 6.99i)T^{2} \) |
| 13 | \( 1 + (-2.98 + 4.94i)T + (-6.06 - 11.4i)T^{2} \) |
| 17 | \( 1 + (-2.50 + 2.04i)T + (3.37 - 16.6i)T^{2} \) |
| 19 | \( 1 + (2.24 + 0.128i)T + (18.8 + 2.16i)T^{2} \) |
| 23 | \( 1 + (1.19 - 8.28i)T + (-22.0 - 6.47i)T^{2} \) |
| 29 | \( 1 + (-1.08 + 2.78i)T + (-21.3 - 19.6i)T^{2} \) |
| 31 | \( 1 + (5.87 + 2.48i)T + (21.6 + 22.2i)T^{2} \) |
| 37 | \( 1 + (0.512 - 5.96i)T + (-36.4 - 6.30i)T^{2} \) |
| 41 | \( 1 + (-2.73 + 1.54i)T + (21.1 - 35.1i)T^{2} \) |
| 43 | \( 1 + (-2.87 + 0.843i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (-0.100 + 0.497i)T + (-43.2 - 18.2i)T^{2} \) |
| 53 | \( 1 + (-12.2 + 6.02i)T + (32.3 - 41.9i)T^{2} \) |
| 59 | \( 1 + (-0.416 - 0.235i)T + (30.4 + 50.5i)T^{2} \) |
| 61 | \( 1 + (11.5 - 1.32i)T + (59.4 - 13.8i)T^{2} \) |
| 67 | \( 1 + (3.04 - 6.67i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (3.09 + 11.7i)T + (-61.8 + 34.9i)T^{2} \) |
| 73 | \( 1 + (-4.79 - 9.08i)T + (-41.2 + 60.2i)T^{2} \) |
| 79 | \( 1 + (-0.0917 - 3.21i)T + (-78.8 + 4.51i)T^{2} \) |
| 83 | \( 1 + (4.59 + 0.795i)T + (78.1 + 27.8i)T^{2} \) |
| 89 | \( 1 + (-8.52 - 5.48i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (1.54 + 2.25i)T + (-35.1 + 90.3i)T^{2} \) |
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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
−10.73917562755687746506286531008, −9.975668512964931368780909784386, −9.024309563047190788684794868747, −8.186742833775298943088014031184, −7.45752513759533306547744871215, −5.80068200844556444540973534721, −5.15479048428182895302497896142, −3.84881119770005794758760692415, −3.43980099998116080557751235154, −2.63044573914253491494599468619,
1.72390106233134540974729135561, 2.66086469919899585822247135726, 3.83942558989535600574832728050, 4.44997561907436018159986189929, 6.04087917484618190847690556223, 6.86795072857781350643434217426, 7.76849754596903042301225030403, 8.787158205232217457242128422407, 9.130418049016373747663180120791, 10.64931318904596693209565529589
