| L(s) = 1 | + (−15.1 − 5.07i)2-s + (66.9 − 21.7i)3-s + (204. + 154. i)4-s + (−7.73 − 624. i)5-s + (−1.12e3 − 9.79i)6-s + 539. i·7-s + (−2.32e3 − 3.37e3i)8-s + (−1.29e3 + 943. i)9-s + (−3.05e3 + 9.52e3i)10-s + (−3.61e3 + 4.98e3i)11-s + (1.70e4 + 5.86e3i)12-s + (2.34e3 − 1.70e3i)13-s + (2.73e3 − 8.18e3i)14-s + (−1.41e4 − 4.16e4i)15-s + (1.80e4 + 6.29e4i)16-s + (−3.96e4 + 1.22e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.948 − 0.317i)2-s + (0.826 − 0.268i)3-s + (0.798 + 0.601i)4-s + (−0.0123 − 0.999i)5-s + (−0.869 − 0.00755i)6-s + 0.224i·7-s + (−0.566 − 0.824i)8-s + (−0.197 + 0.143i)9-s + (−0.305 + 0.952i)10-s + (−0.247 + 0.340i)11-s + (0.821 + 0.282i)12-s + (0.0819 − 0.0595i)13-s + (0.0713 − 0.213i)14-s + (−0.278 − 0.823i)15-s + (0.275 + 0.961i)16-s + (−0.474 + 1.46i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.32760 + 0.346082i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32760 + 0.346082i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (15.1 + 5.07i)T \) |
| 5 | \( 1 + (7.73 + 624. i)T \) |
| good | 3 | \( 1 + (-66.9 + 21.7i)T + (5.30e3 - 3.85e3i)T^{2} \) |
| 7 | \( 1 - 539. iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (3.61e3 - 4.98e3i)T + (-6.62e7 - 2.03e8i)T^{2} \) |
| 13 | \( 1 + (-2.34e3 + 1.70e3i)T + (2.52e8 - 7.75e8i)T^{2} \) |
| 17 | \( 1 + (3.96e4 - 1.22e5i)T + (-5.64e9 - 4.10e9i)T^{2} \) |
| 19 | \( 1 + (-2.18e5 - 7.10e4i)T + (1.37e10 + 9.98e9i)T^{2} \) |
| 23 | \( 1 + (-1.35e5 + 1.86e5i)T + (-2.41e10 - 7.44e10i)T^{2} \) |
| 29 | \( 1 + (-2.27e5 - 7.00e5i)T + (-4.04e11 + 2.94e11i)T^{2} \) |
| 31 | \( 1 + (1.05e6 + 3.43e5i)T + (6.90e11 + 5.01e11i)T^{2} \) |
| 37 | \( 1 + (1.73e6 - 1.25e6i)T + (1.08e12 - 3.34e12i)T^{2} \) |
| 41 | \( 1 + (-2.70e6 + 1.96e6i)T + (2.46e12 - 7.59e12i)T^{2} \) |
| 43 | \( 1 - 3.01e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-2.28e6 + 7.43e5i)T + (1.92e13 - 1.39e13i)T^{2} \) |
| 53 | \( 1 + (3.42e6 + 1.05e7i)T + (-5.03e13 + 3.65e13i)T^{2} \) |
| 59 | \( 1 + (-7.70e6 - 1.06e7i)T + (-4.53e13 + 1.39e14i)T^{2} \) |
| 61 | \( 1 + (-1.03e7 - 7.54e6i)T + (5.92e13 + 1.82e14i)T^{2} \) |
| 67 | \( 1 + (-2.51e7 - 8.17e6i)T + (3.28e14 + 2.38e14i)T^{2} \) |
| 71 | \( 1 + (-6.65e6 + 2.16e6i)T + (5.22e14 - 3.79e14i)T^{2} \) |
| 73 | \( 1 + (-4.14e7 - 3.01e7i)T + (2.49e14 + 7.66e14i)T^{2} \) |
| 79 | \( 1 + (5.38e7 - 1.75e7i)T + (1.22e15 - 8.91e14i)T^{2} \) |
| 83 | \( 1 + (5.49e7 + 1.78e7i)T + (1.82e15 + 1.32e15i)T^{2} \) |
| 89 | \( 1 + (-2.82e7 - 2.05e7i)T + (1.21e15 + 3.74e15i)T^{2} \) |
| 97 | \( 1 + (9.23e6 + 2.84e7i)T + (-6.34e15 + 4.60e15i)T^{2} \) |
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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
−12.43715999772786055922297602272, −11.20661737463224988533982915221, −9.944982808834625767506617054722, −8.849903513346865557872757671099, −8.318833604133827528687942266451, −7.26115367053367941440588587051, −5.52202056177377852336553568965, −3.64782845079890598121076339441, −2.23861825121908795652636492090, −1.16633504344914671605840810443,
0.51590424715637307005190942142, 2.43387445990243032482352774143, 3.32913951319355648831982898506, 5.52782162935029678212283774737, 6.98828002096540969721412949299, 7.70871009907532078676268143845, 9.084506840139613541839020668255, 9.689819153672145888541114381336, 10.97518044403911008816507308744, 11.67480578178169449150549395152
