L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.03 − 0.598i)5-s + (−2.52 + 0.782i)7-s + 0.999·8-s + (1.03 − 0.598i)10-s + (−0.396 − 3.29i)11-s + 3.26i·13-s + (0.586 − 2.57i)14-s + (−0.5 + 0.866i)16-s + (2.21 + 3.82i)17-s + (−6.73 − 3.88i)19-s + 1.19i·20-s + (3.04 + 1.30i)22-s + (5.94 + 3.43i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.463 − 0.267i)5-s + (−0.955 + 0.295i)7-s + 0.353·8-s + (0.327 − 0.189i)10-s + (−0.119 − 0.992i)11-s + 0.904i·13-s + (0.156 − 0.689i)14-s + (−0.125 + 0.216i)16-s + (0.536 + 0.928i)17-s + (−1.54 − 0.892i)19-s + 0.267i·20-s + (0.650 + 0.277i)22-s + (1.24 + 0.715i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.544i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9513739201\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9513739201\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.52 - 0.782i)T \) |
| 11 | \( 1 + (0.396 + 3.29i)T \) |
good | 5 | \( 1 + (1.03 + 0.598i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 - 3.26iT - 13T^{2} \) |
| 17 | \( 1 + (-2.21 - 3.82i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (6.73 + 3.88i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.94 - 3.43i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 7.85T + 29T^{2} \) |
| 31 | \( 1 + (0.103 + 0.179i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.83 + 6.63i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3.15T + 41T^{2} \) |
| 43 | \( 1 - 8.00iT - 43T^{2} \) |
| 47 | \( 1 + (0.355 + 0.205i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.02 + 2.90i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.66 + 2.11i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-9.94 - 5.74i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.61 - 9.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.80iT - 71T^{2} \) |
| 73 | \( 1 + (0.195 - 0.113i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.00 + 1.15i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + (11.4 + 6.58i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−9.421520993588252130366387433498, −8.696060929045236048918519935654, −8.265843401685879375978821282939, −7.11255865806002764615882110328, −6.40724842008477697716809629343, −5.77842203911100924421781651126, −4.59042372515101493653296708100, −3.72055547916917017892066721672, −2.49416541741743349062406782881, −0.73314436195658965289784099080,
0.72908291865487526879093463648, 2.41734755221237295708476171906, 3.24477226205638835819232636577, 4.16453267922512272210951257013, 5.14883686355282530038169631308, 6.44283855602074307332872209313, 7.14792495079851633002769381015, 7.945272555651910192479761611250, 8.767139679086267296847309323516, 9.753071144215017269648697169465