L(s) = 1 | + (−3.95 + 2.28i)2-s + (−4.34 − 7.52i)3-s + (6.40 − 11.0i)4-s + 2.80i·5-s + (34.3 + 19.8i)6-s + (8.28 + 4.78i)7-s + 21.9i·8-s + (−24.2 + 41.9i)9-s + (−6.40 − 11.0i)10-s + (−34.1 + 19.7i)11-s − 111.·12-s − 43.6·14-s + (21.1 − 12.1i)15-s + (1.21 + 2.09i)16-s + (1.00 − 1.74i)17-s − 220. i·18-s + ⋯ |
L(s) = 1 | + (−1.39 + 0.806i)2-s + (−0.835 − 1.44i)3-s + (0.800 − 1.38i)4-s + 0.251i·5-s + (2.33 + 1.34i)6-s + (0.447 + 0.258i)7-s + 0.969i·8-s + (−0.896 + 1.55i)9-s + (−0.202 − 0.350i)10-s + (−0.935 + 0.540i)11-s − 2.67·12-s − 0.832·14-s + (0.363 − 0.209i)15-s + (0.0189 + 0.0327i)16-s + (0.0143 − 0.0248i)17-s − 2.89i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.518027 - 0.0665107i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.518027 - 0.0665107i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + (3.95 - 2.28i)T + (4 - 6.92i)T^{2} \) |
| 3 | \( 1 + (4.34 + 7.52i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 - 2.80iT - 125T^{2} \) |
| 7 | \( 1 + (-8.28 - 4.78i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (34.1 - 19.7i)T + (665.5 - 1.15e3i)T^{2} \) |
| 17 | \( 1 + (-1.00 + 1.74i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-52.1 - 30.0i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-2.23 - 3.87i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (70.3 + 121. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 136. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-160. + 92.8i)T + (2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-268. + 155. i)T + (3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-213. + 370. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 258. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 612.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (448. + 258. i)T + (1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-80.6 + 139. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (43.2 - 24.9i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (242. + 139. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + 467. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 37.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 76.1iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (175. - 101. i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.01e3 - 587. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−12.21150894495402859837263173460, −11.11405181097572827546604374132, −10.27592110092781098090603389472, −8.914823374812782630266780919364, −7.71058854796462197757713278873, −7.38690693007221704056141374540, −6.27718817658486313622108347923, −5.33542760370186146884698174194, −2.09066564144783394977493517517, −0.69386295931389947724831114925,
0.77209761631502330140608579373, 2.97939678783003426053348075906, 4.56556429445112088735192072320, 5.66663184097188512847657350681, 7.59096943961581705063117363655, 8.733786571169526065006014037511, 9.570635485402233662819686949424, 10.41085167416560626859408447713, 11.06631264665241265326989256473, 11.60767287467100065218268632237