| L(s) = 1 | + (1.62 + 2.69i)2-s + (0.331 + 2.98i)3-s + (−2.75 + 5.20i)4-s + (−0.0668 − 0.615i)5-s + (−7.49 + 5.72i)6-s + (−2.85 + 0.963i)7-s + (−5.92 + 0.321i)8-s + (−8.77 + 1.97i)9-s + (1.54 − 1.17i)10-s + (−1.45 − 0.985i)11-s + (−16.4 − 6.49i)12-s + (8.35 + 7.91i)13-s + (−7.23 − 6.14i)14-s + (1.81 − 0.403i)15-s + (2.74 + 4.05i)16-s + (7.09 − 21.0i)17-s + ⋯ |
| L(s) = 1 | + (0.810 + 1.34i)2-s + (0.110 + 0.993i)3-s + (−0.689 + 1.30i)4-s + (−0.0133 − 0.123i)5-s + (−1.24 + 0.954i)6-s + (−0.408 + 0.137i)7-s + (−0.740 + 0.0401i)8-s + (−0.975 + 0.219i)9-s + (0.154 − 0.117i)10-s + (−0.132 − 0.0895i)11-s + (−1.36 − 0.541i)12-s + (0.642 + 0.608i)13-s + (−0.516 − 0.438i)14-s + (0.120 − 0.0268i)15-s + (0.171 + 0.253i)16-s + (0.417 − 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0474260 + 2.11944i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0474260 + 2.11944i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.331 - 2.98i)T \) |
| 59 | \( 1 + (58.1 - 10.0i)T \) |
| good | 2 | \( 1 + (-1.62 - 2.69i)T + (-1.87 + 3.53i)T^{2} \) |
| 5 | \( 1 + (0.0668 + 0.615i)T + (-24.4 + 5.37i)T^{2} \) |
| 7 | \( 1 + (2.85 - 0.963i)T + (39.0 - 29.6i)T^{2} \) |
| 11 | \( 1 + (1.45 + 0.985i)T + (44.7 + 112. i)T^{2} \) |
| 13 | \( 1 + (-8.35 - 7.91i)T + (9.14 + 168. i)T^{2} \) |
| 17 | \( 1 + (-7.09 + 21.0i)T + (-230. - 174. i)T^{2} \) |
| 19 | \( 1 + (-3.53 - 21.5i)T + (-342. + 115. i)T^{2} \) |
| 23 | \( 1 + (-3.35 + 0.932i)T + (453. - 272. i)T^{2} \) |
| 29 | \( 1 + (6.22 - 10.3i)T + (-393. - 743. i)T^{2} \) |
| 31 | \( 1 + (-3.23 + 19.7i)T + (-910. - 306. i)T^{2} \) |
| 37 | \( 1 + (-0.0142 + 0.262i)T + (-1.36e3 - 148. i)T^{2} \) |
| 41 | \( 1 + (-29.9 - 8.31i)T + (1.44e3 + 866. i)T^{2} \) |
| 43 | \( 1 + (-6.65 - 9.81i)T + (-684. + 1.71e3i)T^{2} \) |
| 47 | \( 1 + (-0.778 + 7.15i)T + (-2.15e3 - 474. i)T^{2} \) |
| 53 | \( 1 + (-34.8 + 45.8i)T + (-751. - 2.70e3i)T^{2} \) |
| 61 | \( 1 + (-9.21 + 5.54i)T + (1.74e3 - 3.28e3i)T^{2} \) |
| 67 | \( 1 + (-0.638 - 11.7i)T + (-4.46e3 + 485. i)T^{2} \) |
| 71 | \( 1 + (2.88 - 26.5i)T + (-4.92e3 - 1.08e3i)T^{2} \) |
| 73 | \( 1 + (-48.7 + 57.4i)T + (-862. - 5.25e3i)T^{2} \) |
| 79 | \( 1 + (-43.5 + 109. i)T + (-4.53e3 - 4.29e3i)T^{2} \) |
| 83 | \( 1 + (-31.2 - 67.5i)T + (-4.45e3 + 5.25e3i)T^{2} \) |
| 89 | \( 1 + (-83.2 + 138. i)T + (-3.71e3 - 6.99e3i)T^{2} \) |
| 97 | \( 1 + (63.9 + 75.3i)T + (-1.52e3 + 9.28e3i)T^{2} \) |
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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
−13.35857952831920457750422597226, −12.18277083167650466421951433250, −10.96703697198904939798723148419, −9.729724885563356508593960316520, −8.747840296685868542492425361856, −7.67375985991964267666026421205, −6.38358478087804068401935157710, −5.44904605129119159606457902081, −4.44643063474957475501829514308, −3.29435400903694455240145254835,
1.08972547687316584238607741199, 2.61969612185341003111284178976, 3.64894232089100347125444785401, 5.29089693078230993166236775575, 6.50674075420539889992825179131, 7.81652190741373554119225235970, 9.100606018595257861401694059989, 10.49169202465453774880394523187, 11.12564042069049724069820098550, 12.22642473522484725694552639795
