| L(s) = 1 | + (0.974 − 0.222i)2-s + (−0.875 − 0.483i)3-s + (0.900 − 0.433i)4-s + (0.718 + 1.30i)5-s + (−0.960 − 0.276i)6-s + (−2.43 − 3.05i)7-s + (0.781 − 0.623i)8-s + (0.532 + 0.846i)9-s + (0.989 + 1.10i)10-s + (0.210 − 0.0737i)11-s + (−0.998 − 0.0560i)12-s + (0.576 − 5.11i)13-s + (−3.05 − 2.43i)14-s − 1.48i·15-s + (0.623 − 0.781i)16-s + (3.40 − 2.41i)17-s + ⋯ |
| L(s) = 1 | + (0.689 − 0.157i)2-s + (−0.505 − 0.279i)3-s + (0.450 − 0.216i)4-s + (0.321 + 0.581i)5-s + (−0.392 − 0.113i)6-s + (−0.920 − 1.15i)7-s + (0.276 − 0.220i)8-s + (0.177 + 0.282i)9-s + (0.313 + 0.350i)10-s + (0.0635 − 0.0222i)11-s + (−0.288 − 0.0161i)12-s + (0.159 − 1.41i)13-s + (−0.816 − 0.650i)14-s − 0.383i·15-s + (0.155 − 0.195i)16-s + (0.826 − 0.586i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 678 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0283 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 678 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0283 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.18850 - 1.22272i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18850 - 1.22272i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.974 + 0.222i)T \) |
| 3 | \( 1 + (0.875 + 0.483i)T \) |
| 113 | \( 1 + (7.40 - 7.62i)T \) |
| good | 5 | \( 1 + (-0.718 - 1.30i)T + (-2.66 + 4.23i)T^{2} \) |
| 7 | \( 1 + (2.43 + 3.05i)T + (-1.55 + 6.82i)T^{2} \) |
| 11 | \( 1 + (-0.210 + 0.0737i)T + (8.60 - 6.85i)T^{2} \) |
| 13 | \( 1 + (-0.576 + 5.11i)T + (-12.6 - 2.89i)T^{2} \) |
| 17 | \( 1 + (-3.40 + 2.41i)T + (5.61 - 16.0i)T^{2} \) |
| 19 | \( 1 + (5.07 - 1.46i)T + (16.0 - 10.1i)T^{2} \) |
| 23 | \( 1 + (-1.66 + 5.77i)T + (-19.4 - 12.2i)T^{2} \) |
| 29 | \( 1 + (-4.11 - 5.80i)T + (-9.57 + 27.3i)T^{2} \) |
| 31 | \( 1 + (1.42 + 0.160i)T + (30.2 + 6.89i)T^{2} \) |
| 37 | \( 1 + (-4.53 + 4.05i)T + (4.14 - 36.7i)T^{2} \) |
| 41 | \( 1 + (4.32 + 1.51i)T + (32.0 + 25.5i)T^{2} \) |
| 43 | \( 1 + (0.408 - 2.40i)T + (-40.5 - 14.2i)T^{2} \) |
| 47 | \( 1 + (3.60 - 0.202i)T + (46.7 - 5.26i)T^{2} \) |
| 53 | \( 1 + (2.59 + 5.39i)T + (-33.0 + 41.4i)T^{2} \) |
| 59 | \( 1 + (0.0224 + 0.0779i)T + (-49.9 + 31.3i)T^{2} \) |
| 61 | \( 1 + (-2.32 - 6.64i)T + (-47.6 + 38.0i)T^{2} \) |
| 67 | \( 1 + (0.746 - 13.2i)T + (-66.5 - 7.50i)T^{2} \) |
| 71 | \( 1 + (-2.54 - 6.15i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-2.23 - 0.925i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-1.63 + 1.83i)T + (-8.84 - 78.5i)T^{2} \) |
| 83 | \( 1 + (1.54 + 6.75i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-9.00 + 1.53i)T + (84.0 - 29.3i)T^{2} \) |
| 97 | \( 1 + (-1.69 + 2.13i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−10.43245162440428420337841128698, −9.958582771694792843127282807979, −8.374628312738467242935029651158, −7.25034222396717393340448677764, −6.61881160434882333384669207089, −5.88739204504935372228604657651, −4.76774232993860980253319867062, −3.57996061681240140398095406953, −2.68349388034969970899389384303, −0.78090193054811855300963025552,
1.84408805584061697781547170373, 3.25802154473566936672194277336, 4.40126041594385025813020299867, 5.33101656385657379253394264980, 6.16955819680951575900855019771, 6.69786924201392166578765687688, 8.174637647698025730054001140071, 9.208982144832081357272001182543, 9.651552845723081648348861748900, 10.88985536527179343499551827890
