| L(s) = 1 | + 3·3-s − 2i·5-s + 32i·7-s + 9·9-s + 68i·11-s − 6i·15-s + 14·17-s + 4i·19-s + 96i·21-s − 72·23-s + 121·25-s + 27·27-s + 102·29-s − 136i·31-s + 204i·33-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.178i·5-s + 1.72i·7-s + 0.333·9-s + 1.86i·11-s − 0.103i·15-s + 0.199·17-s + 0.0482i·19-s + 0.997i·21-s − 0.652·23-s + 0.967·25-s + 0.192·27-s + 0.653·29-s − 0.787i·31-s + 1.07i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.307576987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.307576987\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2iT - 125T^{2} \) |
| 7 | \( 1 - 32iT - 343T^{2} \) |
| 11 | \( 1 - 68iT - 1.33e3T^{2} \) |
| 17 | \( 1 - 14T + 4.91e3T^{2} \) |
| 19 | \( 1 - 4iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 72T + 1.21e4T^{2} \) |
| 29 | \( 1 - 102T + 2.43e4T^{2} \) |
| 31 | \( 1 + 136iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 386iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 250iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 140T + 7.95e4T^{2} \) |
| 47 | \( 1 - 296iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 526T + 1.48e5T^{2} \) |
| 59 | \( 1 + 332iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 410T + 2.26e5T^{2} \) |
| 67 | \( 1 - 596iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 880iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 506iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 640T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.38e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.45e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 446iT - 9.12e5T^{2} \) |
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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
−9.124274325822656895203433760919, −8.374353947812951283536540023160, −7.71823997685475186750354013703, −6.75685636226184684741923616383, −5.96066261953728101922559712026, −4.94688989206704124506213762350, −4.39768572170333406033630841599, −2.99794688425899016482321795487, −2.33425926609340014814110981864, −1.48225111171507079653476834222,
0.44891388009551541113233175732, 1.16898147348237885581706773418, 2.64924450538313261881176546796, 3.60408325007648737536345349268, 4.02445682722962573778310429542, 5.23672851518401069917398648869, 6.22415003086038462786123877335, 7.06620956274989615360234939962, 7.64628119322417793580509004451, 8.519926980992739635935754671017
