| L(s) = 1 | + (0.448 − 1.34i)2-s + (0.453 + 0.891i)3-s + (−1.59 − 1.20i)4-s + (2.23 − 0.0568i)5-s + (1.39 − 0.209i)6-s + (2.47 + 2.47i)7-s + (−2.33 + 1.60i)8-s + (−0.587 + 0.809i)9-s + (0.926 − 3.02i)10-s + (2.07 + 2.86i)11-s + (0.347 − 1.96i)12-s + (−0.541 − 3.42i)13-s + (4.43 − 2.21i)14-s + (1.06 + 1.96i)15-s + (1.10 + 3.84i)16-s + (−3.90 − 1.98i)17-s + ⋯ |
| L(s) = 1 | + (0.317 − 0.948i)2-s + (0.262 + 0.514i)3-s + (−0.798 − 0.601i)4-s + (0.999 − 0.0254i)5-s + (0.571 − 0.0853i)6-s + (0.936 + 0.936i)7-s + (−0.824 + 0.566i)8-s + (−0.195 + 0.269i)9-s + (0.293 − 0.956i)10-s + (0.627 + 0.863i)11-s + (0.100 − 0.568i)12-s + (−0.150 − 0.948i)13-s + (1.18 − 0.590i)14-s + (0.275 + 0.507i)15-s + (0.275 + 0.961i)16-s + (−0.946 − 0.482i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 + 0.586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.810 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.74618 - 0.565362i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.74618 - 0.565362i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.448 + 1.34i)T \) |
| 3 | \( 1 + (-0.453 - 0.891i)T \) |
| 5 | \( 1 + (-2.23 + 0.0568i)T \) |
| good | 7 | \( 1 + (-2.47 - 2.47i)T + 7iT^{2} \) |
| 11 | \( 1 + (-2.07 - 2.86i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.541 + 3.42i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (3.90 + 1.98i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (1.81 + 5.58i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.0122 + 0.0770i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (0.807 + 0.262i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.000123 - 4.00e-5i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-5.10 + 0.808i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (4.23 + 3.07i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (4.57 - 4.57i)T - 43iT^{2} \) |
| 47 | \( 1 + (11.5 - 5.88i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (9.59 - 4.88i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (7.12 + 5.17i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-8.17 + 5.93i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (4.18 - 8.21i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (1.99 + 0.647i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-12.5 - 1.99i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-0.609 + 1.87i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-14.4 - 7.37i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (3.49 + 4.80i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (1.72 + 3.38i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−11.47919458328286591922924870074, −10.84742309505510601656198721303, −9.688657379439389609153890741914, −9.214364075451163800542950474193, −8.268093963533916696258440516090, −6.43109129738587813063551605289, −5.14214933773949875289459524308, −4.64347209438409539261207776573, −2.82839284563470108612181710234, −1.89370935462819198411274482375,
1.68041158882545967110984741476, 3.71817882151858132677931194594, 4.85465835326184181902774600731, 6.20703221671945013433397047060, 6.73827871523349564238403856456, 7.979449114899586374922103225428, 8.691245702912635130236233965311, 9.708351173280172766848434912540, 10.94967471752902885415538067620, 12.00932809733663098339092726601
