L(s) = 1 | + (4.19 − 2.42i)2-s + (3.09 + 5.36i)3-s + (7.71 − 13.3i)4-s − 15.2i·5-s + (25.9 + 14.9i)6-s + (−3.73 − 2.15i)7-s − 35.9i·8-s + (−5.69 + 9.87i)9-s + (−36.8 − 63.8i)10-s + (21.2 − 12.2i)11-s + 95.5·12-s − 20.8·14-s + (81.7 − 47.2i)15-s + (−25.2 − 43.8i)16-s + (−63.5 + 110. i)17-s + 55.1i·18-s + ⋯ |
L(s) = 1 | + (1.48 − 0.855i)2-s + (0.596 + 1.03i)3-s + (0.964 − 1.67i)4-s − 1.36i·5-s + (1.76 + 1.02i)6-s + (−0.201 − 0.116i)7-s − 1.58i·8-s + (−0.211 + 0.365i)9-s + (−1.16 − 2.02i)10-s + (0.583 − 0.336i)11-s + 2.29·12-s − 0.398·14-s + (1.40 − 0.812i)15-s + (−0.395 − 0.684i)16-s + (−0.907 + 1.57i)17-s + 0.722i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.905i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.425 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.78351 - 2.40225i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.78351 - 2.40225i\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (-4.19 + 2.42i)T + (4 - 6.92i)T^{2} \) |
| 3 | \( 1 + (-3.09 - 5.36i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 + 15.2iT - 125T^{2} \) |
| 7 | \( 1 + (3.73 + 2.15i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-21.2 + 12.2i)T + (665.5 - 1.15e3i)T^{2} \) |
| 17 | \( 1 + (63.5 - 110. i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-44.8 - 25.8i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-43.6 - 75.6i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (112. + 195. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 108. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (100. - 57.8i)T + (2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (166. - 95.9i)T + (3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (61.6 - 106. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 36.7iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 119.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (696. + 402. i)T + (1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (339. - 587. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (75.7 - 43.7i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-849. - 490. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + 263. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 321.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.04e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (-298. + 172. i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-417. - 241. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−12.33292022593639409439236102004, −11.37285481049453789210017533941, −10.27975757834346458221402638034, −9.316253675002470347362864664947, −8.402030551610510052957251670728, −6.21099107145209731615527606595, −5.02557718690231317729584569105, −4.15799628836929692065344747074, −3.41796898297853641709437278627, −1.55760281620425396050084108774,
2.43671255166385241371821001139, 3.34305938547662240882053110864, 4.92752966065066068153179464506, 6.49375848894455436770265855691, 6.97100446675307927644998763265, 7.60469612095550712446277134288, 9.193023251985338067137422708251, 10.91923834313434836558267546011, 11.96222431722874132357844154056, 12.84990114009127571027972462822