| L(s) = 1 | − 9.07i·5-s + 2.64·7-s − 2.06i·11-s + 4.46·13-s + 9.11i·17-s + 5.96·19-s − 19.7i·23-s − 57.2·25-s − 6.15i·29-s − 53.8·31-s − 23.9i·35-s + 36.4·37-s + 64.4i·41-s − 73.4·43-s + 22.5i·47-s + ⋯ |
| L(s) = 1 | − 1.81i·5-s + 0.377·7-s − 0.188i·11-s + 0.343·13-s + 0.536i·17-s + 0.314·19-s − 0.856i·23-s − 2.29·25-s − 0.212i·29-s − 1.73·31-s − 0.685i·35-s + 0.984·37-s + 1.57i·41-s − 1.70·43-s + 0.478i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2520600986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2520600986\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
| good | 5 | \( 1 + 9.07iT - 25T^{2} \) |
| 11 | \( 1 + 2.06iT - 121T^{2} \) |
| 13 | \( 1 - 4.46T + 169T^{2} \) |
| 17 | \( 1 - 9.11iT - 289T^{2} \) |
| 19 | \( 1 - 5.96T + 361T^{2} \) |
| 23 | \( 1 + 19.7iT - 529T^{2} \) |
| 29 | \( 1 + 6.15iT - 841T^{2} \) |
| 31 | \( 1 + 53.8T + 961T^{2} \) |
| 37 | \( 1 - 36.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 64.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 73.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 22.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 14.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 71.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 38.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 92.4T + 4.48e3T^{2} \) |
| 71 | \( 1 + 29.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 62.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 120.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 110. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 125. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 89.6T + 9.40e3T^{2} \) |
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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
−8.062271720506108779289432865051, −7.21139254838676224567448886615, −6.03015577857333115148074237363, −5.58798054854652565304057812010, −4.61925435015456652236720369031, −4.29327914529146489474232192491, −3.15134292360954436572375974356, −1.79018327278306906048935444230, −1.16324710083267696075217577846, −0.05135009309011323468070128093,
1.62670797877387145325411480706, 2.48883920387043821521334298136, 3.37059815935791851533475057025, 3.91361737447834648186467913299, 5.18756554804560358870430345921, 5.82487414058280962829022881094, 6.75638798549282606826236141173, 7.21344372467552162335461381896, 7.77250256055169532606346593417, 8.724412616325721321895643677957
