| L(s) = 1 | + (1 + 2.82i)3-s − 6·7-s + (−7.00 + 5.65i)9-s + 5.65i·11-s − 10·13-s − 22.6i·17-s − 2·19-s + (−6 − 16.9i)21-s + 11.3i·23-s + (−23.0 − 14.1i)27-s − 16.9i·29-s + 22·31-s + (−16.0 + 5.65i)33-s + 6·37-s + (−10 − 28.2i)39-s + ⋯ |
| L(s) = 1 | + (0.333 + 0.942i)3-s − 0.857·7-s + (−0.777 + 0.628i)9-s + 0.514i·11-s − 0.769·13-s − 1.33i·17-s − 0.105·19-s + (−0.285 − 0.808i)21-s + 0.491i·23-s + (−0.851 − 0.523i)27-s − 0.585i·29-s + 0.709·31-s + (−0.484 + 0.171i)33-s + 0.162·37-s + (−0.256 − 0.725i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7436960828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7436960828\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 - 2.82i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 6T + 49T^{2} \) |
| 11 | \( 1 - 5.65iT - 121T^{2} \) |
| 13 | \( 1 + 10T + 169T^{2} \) |
| 17 | \( 1 + 22.6iT - 289T^{2} \) |
| 19 | \( 1 + 2T + 361T^{2} \) |
| 23 | \( 1 - 11.3iT - 529T^{2} \) |
| 29 | \( 1 + 16.9iT - 841T^{2} \) |
| 31 | \( 1 - 22T + 961T^{2} \) |
| 37 | \( 1 - 6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 33.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 82T + 1.84e3T^{2} \) |
| 47 | \( 1 + 67.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 62.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 73.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 86T + 3.72e3T^{2} \) |
| 67 | \( 1 - 2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 124. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 82T + 5.32e3T^{2} \) |
| 79 | \( 1 + 10T + 6.24e3T^{2} \) |
| 83 | \( 1 - 73.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 33.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 94T + 9.40e3T^{2} \) |
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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
−9.539496706816842778479241526851, −8.865899474048917224386818302490, −7.73208241582838318735016481703, −7.03060300936997319669046008600, −5.89430904814749831712389516501, −4.98978719922766794400831164196, −4.19123105092579690269786871711, −3.12973024380928673156128596219, −2.33266767085190593101745319834, −0.22025152440695377967663022553,
1.16352996626922221150009811631, 2.50217090886847594359826697842, 3.28274254962164783730904903958, 4.46104698936827657481671950480, 5.93036831131076501388227391352, 6.30207585723588457685137867977, 7.28710177058315341499725494517, 8.028910012712114874943989927378, 8.830477909789199560914017218736, 9.571835285496295059046264538994
