| L(s) = 1 | + (−2.03 − 0.907i)2-s + (1.68 − 1.22i)3-s + (1.99 + 2.21i)4-s + (−2.26 + 0.480i)5-s + (−4.55 + 0.967i)6-s + (−0.0297 + 0.282i)7-s + (−0.677 − 2.08i)8-s + (0.416 − 1.28i)9-s + (5.04 + 1.07i)10-s − 11-s + (6.08 + 1.29i)12-s + (−1.47 − 2.56i)13-s + (0.317 − 0.549i)14-s + (−3.22 + 3.58i)15-s + (0.112 − 1.06i)16-s + (−1.89 − 2.10i)17-s + ⋯ |
| L(s) = 1 | + (−1.44 − 0.641i)2-s + (0.973 − 0.707i)3-s + (0.997 + 1.10i)4-s + (−1.01 + 0.214i)5-s + (−1.85 + 0.394i)6-s + (−0.0112 + 0.106i)7-s + (−0.239 − 0.736i)8-s + (0.138 − 0.427i)9-s + (1.59 + 0.339i)10-s − 0.301·11-s + (1.75 + 0.373i)12-s + (−0.410 − 0.710i)13-s + (0.0848 − 0.146i)14-s + (−0.832 + 0.924i)15-s + (0.0280 − 0.266i)16-s + (−0.459 − 0.510i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0336 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0336 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.124690 + 0.120563i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.124690 + 0.120563i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 61 | \( 1 + (2.36 - 7.44i)T \) |
| good | 2 | \( 1 + (2.03 + 0.907i)T + (1.33 + 1.48i)T^{2} \) |
| 3 | \( 1 + (-1.68 + 1.22i)T + (0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 + (2.26 - 0.480i)T + (4.56 - 2.03i)T^{2} \) |
| 7 | \( 1 + (0.0297 - 0.282i)T + (-6.84 - 1.45i)T^{2} \) |
| 13 | \( 1 + (1.47 + 2.56i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.89 + 2.10i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.0603 + 0.574i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (2.66 - 8.21i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (2.38 - 4.13i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.57 - 1.59i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-2.36 - 1.72i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (6.73 + 4.89i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-0.235 + 0.261i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (5.39 - 9.34i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.44 + 7.51i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (0.643 + 0.286i)T + (39.4 + 43.8i)T^{2} \) |
| 67 | \( 1 + (2.59 - 0.551i)T + (61.2 - 27.2i)T^{2} \) |
| 71 | \( 1 + (-5.63 - 1.19i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (0.596 + 0.126i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-4.02 + 4.46i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (7.20 + 3.20i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (3.37 - 2.45i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (6.53 - 2.91i)T + (64.9 - 72.0i)T^{2} \) |
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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
−10.67550460890837002614478208775, −9.658498970726364834089596121655, −8.936336632622422413253002514307, −8.112911684861246992363380722047, −7.59717262170963647326821864551, −7.09401730068650962642462444022, −5.26853354323230360856809189444, −3.53161410662561856381485041195, −2.72453400154599406009826330720, −1.60016939594862547279114553658,
0.12751534232376061037461418644, 2.20737431272150242045951653176, 3.77910527168887259976754945824, 4.49199434642972595606719348947, 6.21600915262919332334158342295, 7.16463715267887808493081908738, 8.137062612182201463655050129811, 8.405168665139512894403523456495, 9.272469270142081551895180326840, 9.948480594346475200409003672893
