L(s) = 1 | + (−3.89 − 0.889i)2-s + 5.19i·3-s + (14.4 + 6.93i)4-s + (4.61 − 20.2i)6-s − 44.0i·7-s + (−50.0 − 39.8i)8-s − 27·9-s + 81.7i·11-s + (−36.0 + 74.9i)12-s − 192.·13-s + (−39.2 + 171. i)14-s + (159. + 199. i)16-s + 415.·17-s + (105. + 24.0i)18-s + 23.6i·19-s + ⋯ |
L(s) = 1 | + (−0.974 − 0.222i)2-s + 0.577i·3-s + (0.901 + 0.433i)4-s + (0.128 − 0.562i)6-s − 0.899i·7-s + (−0.782 − 0.622i)8-s − 0.333·9-s + 0.675i·11-s + (−0.250 + 0.520i)12-s − 1.13·13-s + (−0.200 + 0.877i)14-s + (0.624 + 0.781i)16-s + 1.43·17-s + (0.324 + 0.0740i)18-s + 0.0655i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.9783104425\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9783104425\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (3.89 + 0.889i)T \) |
| 3 | \( 1 - 5.19iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 44.0iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 81.7iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 192.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 415.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 23.6iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 525. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 254.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.30e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.22e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.95e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.49e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.00e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 20.8T + 7.89e6T^{2} \) |
| 59 | \( 1 + 2.21e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.19e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 6.56e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 8.34e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 6.09e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 6.12e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 9.94e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 6.74e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 1.53e3T + 8.85e7T^{2} \) |
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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
−10.91997336519731917400289044088, −10.10031653147088352883637992805, −9.742069058793639664150128282936, −8.481942031408768582510653364713, −7.52389969735145924277781743361, −6.74305484377038540015482918324, −5.16587061259889378627224702941, −3.87139093490737236932632979344, −2.60034432463964419587495102368, −0.981412996437255558718367055419,
0.51233149593907454224028838611, 1.96566203697345122118289471072, 3.08810159530601688729399841415, 5.37813087528287924340733455395, 6.04789166475186015785888610711, 7.35202867341125343586445739527, 7.963037931384332122339913879009, 9.037897960592608889072923784929, 9.757714301995020317340005379736, 10.87917197043908678579791560885