| L(s) = 1 | + (0.474 − 0.699i)2-s + (−3.34 + 3.97i)3-s + (2.69 + 6.76i)4-s + (−7.58 + 16.3i)5-s + (1.19 + 4.22i)6-s + (7.28 + 26.2i)7-s + (12.6 + 2.77i)8-s + (−4.67 − 26.5i)9-s + (7.86 + 13.0i)10-s + (−13.5 − 12.8i)11-s + (−35.9 − 11.8i)12-s + (7.39 − 68.0i)13-s + (21.8 + 7.34i)14-s + (−39.8 − 84.9i)15-s + (−34.3 + 32.5i)16-s + (−17.6 − 4.88i)17-s + ⋯ |
| L(s) = 1 | + (0.167 − 0.247i)2-s + (−0.643 + 0.765i)3-s + (0.337 + 0.846i)4-s + (−0.678 + 1.46i)5-s + (0.0815 + 0.287i)6-s + (0.393 + 1.41i)7-s + (0.557 + 0.122i)8-s + (−0.173 − 0.984i)9-s + (0.248 + 0.413i)10-s + (−0.372 − 0.353i)11-s + (−0.864 − 0.285i)12-s + (0.157 − 1.45i)13-s + (0.416 + 0.140i)14-s + (−0.686 − 1.46i)15-s + (−0.537 + 0.509i)16-s + (−0.251 − 0.0697i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0490i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0306222 + 1.24705i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0306222 + 1.24705i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.34 - 3.97i)T \) |
| 59 | \( 1 + (89.0 - 444. i)T \) |
| good | 2 | \( 1 + (-0.474 + 0.699i)T + (-2.96 - 7.43i)T^{2} \) |
| 5 | \( 1 + (7.58 - 16.3i)T + (-80.9 - 95.2i)T^{2} \) |
| 7 | \( 1 + (-7.28 - 26.2i)T + (-293. + 176. i)T^{2} \) |
| 11 | \( 1 + (13.5 + 12.8i)T + (72.0 + 1.32e3i)T^{2} \) |
| 13 | \( 1 + (-7.39 + 68.0i)T + (-2.14e3 - 472. i)T^{2} \) |
| 17 | \( 1 + (17.6 + 4.88i)T + (4.20e3 + 2.53e3i)T^{2} \) |
| 19 | \( 1 + (-87.6 - 66.6i)T + (1.83e3 + 6.60e3i)T^{2} \) |
| 23 | \( 1 + (-47.0 - 88.7i)T + (-6.82e3 + 1.00e4i)T^{2} \) |
| 29 | \( 1 + (-194. + 131. i)T + (9.02e3 - 2.26e4i)T^{2} \) |
| 31 | \( 1 + (-44.0 - 57.9i)T + (-7.96e3 + 2.87e4i)T^{2} \) |
| 37 | \( 1 + (-40.4 - 183. i)T + (-4.59e4 + 2.12e4i)T^{2} \) |
| 41 | \( 1 + (218. + 115. i)T + (3.86e4 + 5.70e4i)T^{2} \) |
| 43 | \( 1 + (314. + 331. i)T + (-4.30e3 + 7.93e4i)T^{2} \) |
| 47 | \( 1 + (55.8 - 25.8i)T + (6.72e4 - 7.91e4i)T^{2} \) |
| 53 | \( 1 + (178. - 296. i)T + (-6.97e4 - 1.31e5i)T^{2} \) |
| 61 | \( 1 + (-105. - 71.3i)T + (8.40e4 + 2.10e5i)T^{2} \) |
| 67 | \( 1 + (-8.02 + 36.4i)T + (-2.72e5 - 1.26e5i)T^{2} \) |
| 71 | \( 1 + (67.3 + 145. i)T + (-2.31e5 + 2.72e5i)T^{2} \) |
| 73 | \( 1 + (-160. + 477. i)T + (-3.09e5 - 2.35e5i)T^{2} \) |
| 79 | \( 1 + (-56.2 + 1.03e3i)T + (-4.90e5 - 5.33e4i)T^{2} \) |
| 83 | \( 1 + (150. - 916. i)T + (-5.41e5 - 1.82e5i)T^{2} \) |
| 89 | \( 1 + (-427. - 629. i)T + (-2.60e5 + 6.54e5i)T^{2} \) |
| 97 | \( 1 + (-162. - 482. i)T + (-7.26e5 + 5.52e5i)T^{2} \) |
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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
−12.06698009538770612474615543723, −11.84060798380603416887407537760, −10.90054604095084321047737571356, −10.15282683883270039383956428776, −8.515919084409512333806420145224, −7.62844628019618979881418752669, −6.29148102062889774951856056039, −5.17025906460908853195337104850, −3.44483929965815307600180928525, −2.85672935553803647847399206103,
0.61570189229910379538279124355, 1.51709096686088679397818777047, 4.60187394807455785014310994548, 4.91400990133795232823223792352, 6.62496186339062014445884936979, 7.31510612319415546655922153405, 8.440138894252766476906796707755, 9.848258072758859148913537756919, 11.08684382342825786198947629490, 11.63105223073003496116792424134
