| L(s) = 1 | + (−1.22 − 1.36i)2-s + (−2.00 − 0.894i)3-s + (−0.142 + 1.35i)4-s + (0.621 + 0.132i)5-s + (1.24 + 3.83i)6-s + (1.80 − 1.93i)7-s + (−0.943 + 0.685i)8-s + (1.22 + 1.36i)9-s + (−0.582 − 1.00i)10-s + (1.49 − 2.59i)12-s + (0.556 − 1.71i)13-s + (−4.85 − 0.0760i)14-s + (−1.13 − 0.821i)15-s + (4.76 + 1.01i)16-s + (1.89 − 2.10i)17-s + (0.351 − 3.34i)18-s + ⋯ |
| L(s) = 1 | + (−0.867 − 0.963i)2-s + (−1.15 − 0.516i)3-s + (−0.0713 + 0.678i)4-s + (0.278 + 0.0590i)5-s + (0.508 + 1.56i)6-s + (0.680 − 0.732i)7-s + (−0.333 + 0.242i)8-s + (0.409 + 0.454i)9-s + (−0.184 − 0.319i)10-s + (0.433 − 0.749i)12-s + (0.154 − 0.475i)13-s + (−1.29 − 0.0203i)14-s + (−0.291 − 0.212i)15-s + (1.19 + 0.252i)16-s + (0.459 − 0.510i)17-s + (0.0828 − 0.788i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.198752 + 0.298584i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.198752 + 0.298584i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-1.80 + 1.93i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (1.22 + 1.36i)T + (-0.209 + 1.98i)T^{2} \) |
| 3 | \( 1 + (2.00 + 0.894i)T + (2.00 + 2.22i)T^{2} \) |
| 5 | \( 1 + (-0.621 - 0.132i)T + (4.56 + 2.03i)T^{2} \) |
| 13 | \( 1 + (-0.556 + 1.71i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.89 + 2.10i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.581 + 5.53i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (1.08 - 1.87i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (8.43 + 6.13i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (6.28 - 1.33i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (-5.54 + 2.46i)T + (24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (-6.09 + 4.42i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 4.86T + 43T^{2} \) |
| 47 | \( 1 + (-0.296 - 2.81i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (7.30 - 1.55i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (1.23 - 11.7i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (4.23 + 0.900i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-0.801 - 1.38i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.32 - 4.08i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (1.67 - 15.9i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (3.18 + 3.53i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-2.85 - 8.77i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (0.182 - 0.315i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.802 - 2.46i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−9.749783872677533013401800076669, −9.097658778120302600900245151643, −7.84200946907040592800521639283, −7.25298571073665951511520124038, −5.94856456807560911958642321180, −5.41598792924408460326382706348, −4.02756899852715183262993129478, −2.50300668331330301094520807504, −1.29659477782854948752086876552, −0.29666375548431820589050022939,
1.71261848047443746771631929749, 3.70688415396699934978173496329, 4.98281833184488902022142469727, 5.95726147467008000828128166812, 6.07283153628506366604609787936, 7.52174841928858112155329024751, 8.121299900819969036157623929866, 9.147570187961329820368899976499, 9.696565649758620133080877447925, 10.70320517896388430556237161462
