L(s) = 1 | − 2-s + (0.869 − 1.49i)3-s + 4-s − 1.66i·5-s + (−0.869 + 1.49i)6-s − 3.57·7-s − 8-s + (−1.48 − 2.60i)9-s + 1.66i·10-s − 4.82·11-s + (0.869 − 1.49i)12-s + 0.418i·13-s + 3.57·14-s + (−2.48 − 1.44i)15-s + 16-s + 0.742i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.502 − 0.864i)3-s + 0.5·4-s − 0.742i·5-s + (−0.355 + 0.611i)6-s − 1.35·7-s − 0.353·8-s + (−0.495 − 0.868i)9-s + 0.524i·10-s − 1.45·11-s + (0.251 − 0.432i)12-s + 0.116i·13-s + 0.956·14-s + (−0.642 − 0.372i)15-s + 0.250·16-s + 0.180i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.264i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0779952 - 0.579371i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0779952 - 0.579371i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.869 + 1.49i)T \) |
| 59 | \( 1 + (-5.47 - 5.38i)T \) |
good | 5 | \( 1 + 1.66iT - 5T^{2} \) |
| 7 | \( 1 + 3.57T + 7T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 0.418iT - 13T^{2} \) |
| 17 | \( 1 - 0.742iT - 17T^{2} \) |
| 19 | \( 1 - 1.24T + 19T^{2} \) |
| 23 | \( 1 - 1.25T + 23T^{2} \) |
| 29 | \( 1 - 4.78iT - 29T^{2} \) |
| 31 | \( 1 + 10.4iT - 31T^{2} \) |
| 37 | \( 1 + 6.02iT - 37T^{2} \) |
| 41 | \( 1 + 9.24iT - 41T^{2} \) |
| 43 | \( 1 - 3.58iT - 43T^{2} \) |
| 47 | \( 1 + 8.39T + 47T^{2} \) |
| 53 | \( 1 + 9.66iT - 53T^{2} \) |
| 61 | \( 1 + 9.08iT - 61T^{2} \) |
| 67 | \( 1 + 13.0iT - 67T^{2} \) |
| 71 | \( 1 - 4.01iT - 71T^{2} \) |
| 73 | \( 1 - 10.6iT - 73T^{2} \) |
| 79 | \( 1 - 3.55T + 79T^{2} \) |
| 83 | \( 1 + 2.87T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 - 11.5iT - 97T^{2} \) |
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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
−10.95581581726219086221310924872, −9.831905273571093651721480532359, −9.151214928524464038968547652122, −8.252311428157221318248102208910, −7.42237729760466416561537149257, −6.47320942826670643312044386056, −5.38803806778467572010911440168, −3.43316960280639529696700690842, −2.28211974081840585064491615231, −0.44038283552706877865809875496,
2.75059940896852471278931911046, 3.23803952970376798417246378690, 4.99580644492729200764942043894, 6.27860994755856958010651042599, 7.31573079607611625570094090941, 8.296828760314077096337823107205, 9.280679739517931336114181371808, 10.25663725548389229507880189388, 10.37770811813735534936973403293, 11.54385675600396742251036226615