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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 + 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.743 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.439484 - 1.14515i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.439484 - 1.14515i\) |
\(L(\frac{3}{2})\) |
|
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
−10.09884777904116196027989905955, −9.659014763081583606326164397058, −8.164505611175389102210993583691, −6.97354921336916914005853754635, −6.08379836217135695224938570289, −5.08513477749116290176538909816, −4.52493791812233687743411488031, −3.50599990289393268823889561546, −2.38355122103453376438042953636, −0.51834666139145632360349737722,
1.64125291551698472777931867594, 3.42010822153434721912453863981, 4.66210031859434826482530750152, 5.58431767624240545850546114455, 6.14029485622512255736959095940, 6.59975469202007229172223133384, 7.51787245276340108735570604448, 8.635613287128935208538445320111, 9.795058130191064107523759387692, 10.54146804219173386346865643970