L(s) = 1 | + (0.889 + 3.89i)2-s + 5.19·3-s + (−14.4 + 6.93i)4-s + (4.61 + 20.2i)6-s + 44.0·7-s + (−39.8 − 50.0i)8-s + 27·9-s − 81.7i·11-s + (−74.9 + 36.0i)12-s − 192. i·13-s + (39.2 + 171. i)14-s + (159. − 199. i)16-s − 415. i·17-s + (24.0 + 105. i)18-s + 23.6i·19-s + ⋯ |
L(s) = 1 | + (0.222 + 0.974i)2-s + 0.577·3-s + (−0.901 + 0.433i)4-s + (0.128 + 0.562i)6-s + 0.899·7-s + (−0.622 − 0.782i)8-s + 0.333·9-s − 0.675i·11-s + (−0.520 + 0.250i)12-s − 1.13i·13-s + (0.200 + 0.877i)14-s + (0.624 − 0.781i)16-s − 1.43i·17-s + (0.0740 + 0.324i)18-s + 0.0655i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0153i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.999 + 0.0153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.374535435\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.374535435\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.889 - 3.89i)T \) |
| 3 | \( 1 - 5.19T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 44.0T + 2.40e3T^{2} \) |
| 11 | \( 1 + 81.7iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 192. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 415. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 23.6iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 525.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 254.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.30e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.22e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.95e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.49e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.00e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 20.8iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 2.21e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.19e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 6.56e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 8.34e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 6.09e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 6.12e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 9.94e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 6.74e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 1.53e3iT - 8.85e7T^{2} \) |
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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
−11.13498071590888185567976905795, −9.861183286438137987786489502755, −8.951475556125906881168090765134, −7.938851078727209784936568557830, −7.56149948729560282570485448225, −6.08464433306232605226106766977, −5.15124854590643523850142717842, −4.02913450460899595064522755435, −2.72512389419101600145813731911, −0.66491607538276446393868775179,
1.48032237335619567755599106645, 2.26325205776344366958161433095, 3.89016856411536500162433918115, 4.55221554936107978054310885013, 5.92996319943846369404576107852, 7.45915125136558070158506123716, 8.526469363695402856631586981074, 9.251497244478983945637150364712, 10.31535181127301920492710464850, 11.06137650646370250657390108361