| L(s) = 1 | + (−1.27 − 0.615i)2-s + (−0.279 + 0.599i)3-s + (1.24 + 1.56i)4-s + (1.95 − 1.09i)5-s + (0.724 − 0.590i)6-s + (−0.490 + 1.82i)7-s + (−0.614 − 2.76i)8-s + (1.64 + 1.96i)9-s + (−3.15 + 0.189i)10-s + (1.54 + 0.889i)11-s + (−1.28 + 0.305i)12-s + (−1.03 − 2.22i)13-s + (1.75 − 2.02i)14-s + (0.109 + 1.47i)15-s + (−0.917 + 3.89i)16-s + (−0.526 + 0.0460i)17-s + ⋯ |
| L(s) = 1 | + (−0.900 − 0.435i)2-s + (−0.161 + 0.345i)3-s + (0.620 + 0.784i)4-s + (0.872 − 0.488i)5-s + (0.295 − 0.241i)6-s + (−0.185 + 0.691i)7-s + (−0.217 − 0.976i)8-s + (0.549 + 0.654i)9-s + (−0.998 + 0.0598i)10-s + (0.464 + 0.268i)11-s + (−0.371 + 0.0882i)12-s + (−0.288 − 0.618i)13-s + (0.467 − 0.541i)14-s + (0.0282 + 0.380i)15-s + (−0.229 + 0.973i)16-s + (−0.127 + 0.0111i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.991491 + 0.206176i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.991491 + 0.206176i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.27 + 0.615i)T \) |
| 5 | \( 1 + (-1.95 + 1.09i)T \) |
| 19 | \( 1 + (3.49 - 2.60i)T \) |
| good | 3 | \( 1 + (0.279 - 0.599i)T + (-1.92 - 2.29i)T^{2} \) |
| 7 | \( 1 + (0.490 - 1.82i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.54 - 0.889i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.03 + 2.22i)T + (-8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (0.526 - 0.0460i)T + (16.7 - 2.95i)T^{2} \) |
| 23 | \( 1 + (-4.62 - 3.23i)T + (7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-5.17 - 6.16i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-3.36 + 1.94i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.166 + 0.166i)T - 37iT^{2} \) |
| 41 | \( 1 + (-6.13 + 2.23i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.61 - 2.30i)T + (-14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (-1.04 - 0.0913i)T + (46.2 + 8.16i)T^{2} \) |
| 53 | \( 1 + (1.56 - 2.24i)T + (-18.1 - 49.8i)T^{2} \) |
| 59 | \( 1 + (7.66 + 6.43i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.16 + 6.57i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.76 - 0.242i)T + (65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (-1.50 - 0.264i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (8.80 + 4.10i)T + (46.9 + 55.9i)T^{2} \) |
| 79 | \( 1 + (-0.721 + 0.262i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (8.75 + 2.34i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (3.72 - 10.2i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (13.2 - 1.15i)T + (95.5 - 16.8i)T^{2} \) |
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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
−11.10958916525406377691180227536, −10.36210759723585335829410642511, −9.604076012144795934376811428785, −8.929917360386068504594589197603, −7.954745941033878204903816736345, −6.76273927358837843918733718079, −5.65031145047645039199887107968, −4.45321931950407344118747515519, −2.78055813899536961173024529505, −1.54770102958789298387439222513,
1.05685487616581650441466920082, 2.56179253671872590302339077355, 4.44891347352084126636515558240, 6.05937501145500409079065983478, 6.69004511777116000463920890280, 7.24826993836349738860032687480, 8.675994632974575147442164738065, 9.472672216284994121416394013692, 10.23897155724570734759591384086, 11.00040150345210384616190001385
