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
−9.753071144215017269648697169465, −8.767139679086267296847309323516, −7.945272555651910192479761611250, −7.14792495079851633002769381015, −6.44283855602074307332872209313, −5.14883686355282530038169631308, −4.16453267922512272210951257013, −3.24477226205638835819232636577, −2.41734755221237295708476171906, −0.72908291865487526879093463648,
0.73314436195658965289784099080, 2.49416541741743349062406782881, 3.72055547916917017892066721672, 4.59042372515101493653296708100, 5.77842203911100924421781651126, 6.40724842008477697716809629343, 7.11255865806002764615882110328, 8.265843401685879375978821282939, 8.696060929045236048918519935654, 9.421520993588252130366387433498