| L(s) = 1 | + (0.347 + 0.143i)2-s + (−2.84 − 14.2i)3-s + (−11.2 − 11.2i)4-s + (33.0 + 22.0i)5-s + (1.06 − 5.37i)6-s + (6.11 − 4.08i)7-s + (−4.58 − 11.0i)8-s + (−121. + 50.1i)9-s + (8.30 + 12.4i)10-s + (147. + 29.4i)11-s + (−128. + 191. i)12-s + (32.4 − 32.4i)13-s + (2.71 − 0.539i)14-s + (221. − 534. i)15-s + 249. i·16-s + (−287. + 25.2i)17-s + ⋯ |
| L(s) = 1 | + (0.0869 + 0.0359i)2-s + (−0.315 − 1.58i)3-s + (−0.700 − 0.700i)4-s + (1.32 + 0.883i)5-s + (0.0296 − 0.149i)6-s + (0.124 − 0.0833i)7-s + (−0.0716 − 0.173i)8-s + (−1.49 + 0.618i)9-s + (0.0830 + 0.124i)10-s + (1.22 + 0.243i)11-s + (−0.890 + 1.33i)12-s + (0.192 − 0.192i)13-s + (0.0138 − 0.00275i)14-s + (0.984 − 2.37i)15-s + 0.973i·16-s + (−0.996 + 0.0872i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.933362 - 0.742311i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.933362 - 0.742311i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (287. - 25.2i)T \) |
| good | 2 | \( 1 + (-0.347 - 0.143i)T + (11.3 + 11.3i)T^{2} \) |
| 3 | \( 1 + (2.84 + 14.2i)T + (-74.8 + 30.9i)T^{2} \) |
| 5 | \( 1 + (-33.0 - 22.0i)T + (239. + 577. i)T^{2} \) |
| 7 | \( 1 + (-6.11 + 4.08i)T + (918. - 2.21e3i)T^{2} \) |
| 11 | \( 1 + (-147. - 29.4i)T + (1.35e4 + 5.60e3i)T^{2} \) |
| 13 | \( 1 + (-32.4 + 32.4i)T - 2.85e4iT^{2} \) |
| 19 | \( 1 + (-274. - 113. i)T + (9.21e4 + 9.21e4i)T^{2} \) |
| 23 | \( 1 + (-60.6 + 304. i)T + (-2.58e5 - 1.07e5i)T^{2} \) |
| 29 | \( 1 + (420. - 629. i)T + (-2.70e5 - 6.53e5i)T^{2} \) |
| 31 | \( 1 + (1.55e3 - 309. i)T + (8.53e5 - 3.53e5i)T^{2} \) |
| 37 | \( 1 + (-190. - 957. i)T + (-1.73e6 + 7.17e5i)T^{2} \) |
| 41 | \( 1 + (-1.33e3 + 891. i)T + (1.08e6 - 2.61e6i)T^{2} \) |
| 43 | \( 1 + (-1.27e3 + 526. i)T + (2.41e6 - 2.41e6i)T^{2} \) |
| 47 | \( 1 + (-236. + 236. i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (1.05e3 + 436. i)T + (5.57e6 + 5.57e6i)T^{2} \) |
| 59 | \( 1 + (-435. - 1.05e3i)T + (-8.56e6 + 8.56e6i)T^{2} \) |
| 61 | \( 1 + (1.10e3 + 1.66e3i)T + (-5.29e6 + 1.27e7i)T^{2} \) |
| 67 | \( 1 + 3.67e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + (489. + 2.46e3i)T + (-2.34e7 + 9.72e6i)T^{2} \) |
| 73 | \( 1 + (-3.62e3 - 2.42e3i)T + (1.08e7 + 2.62e7i)T^{2} \) |
| 79 | \( 1 + (9.23e3 + 1.83e3i)T + (3.59e7 + 1.49e7i)T^{2} \) |
| 83 | \( 1 + (278. - 673. i)T + (-3.35e7 - 3.35e7i)T^{2} \) |
| 89 | \( 1 + (666. + 666. i)T + 6.27e7iT^{2} \) |
| 97 | \( 1 + (-861. + 1.28e3i)T + (-3.38e7 - 8.17e7i)T^{2} \) |
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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
−18.04374842076971058162302282896, −17.29823371717418687473028536655, −14.52295961983757103900372006313, −13.86585918742918435856068675853, −12.76893635512877948659012182485, −10.91923920431963804421911370820, −9.238541476690968940729708250030, −6.88370279130735540836368461449, −5.86831686913914023312075255702, −1.61852100211353573089981938422,
4.14777471627972710112974963540, 5.51327941331524631977928157792, 9.104815094905047221800621731811, 9.432291136366914874157099806369, 11.40394288927792493056957265830, 13.17297446724501408248826601697, 14.38987726440335919279413382685, 16.18127245796216491685654313808, 17.03310239143966181110483418061, 17.79238211646849443904781267253
