| L(s) = 1 | + (−2.10 − 1.89i)2-s + (1.94 + 4.68i)3-s + (0.829 + 7.95i)4-s + (4.93 + 2.04i)5-s + (4.79 − 13.5i)6-s + (14.0 + 14.0i)7-s + (13.3 − 18.2i)8-s + (0.893 − 0.893i)9-s + (−6.49 − 13.6i)10-s + (−3.78 + 9.14i)11-s + (−35.6 + 19.3i)12-s + (−64.7 + 26.8i)13-s + (−2.92 − 56.2i)14-s + 27.0i·15-s + (−62.6 + 13.2i)16-s − 79.3i·17-s + ⋯ |
| L(s) = 1 | + (−0.742 − 0.669i)2-s + (0.373 + 0.901i)3-s + (0.103 + 0.994i)4-s + (0.441 + 0.182i)5-s + (0.326 − 0.920i)6-s + (0.760 + 0.760i)7-s + (0.588 − 0.808i)8-s + (0.0331 − 0.0331i)9-s + (−0.205 − 0.431i)10-s + (−0.103 + 0.250i)11-s + (−0.858 + 0.465i)12-s + (−1.38 + 0.572i)13-s + (−0.0558 − 1.07i)14-s + 0.466i·15-s + (−0.978 + 0.206i)16-s − 1.13i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.03124 + 0.239355i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03124 + 0.239355i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.10 + 1.89i)T \) |
| good | 3 | \( 1 + (-1.94 - 4.68i)T + (-19.0 + 19.0i)T^{2} \) |
| 5 | \( 1 + (-4.93 - 2.04i)T + (88.3 + 88.3i)T^{2} \) |
| 7 | \( 1 + (-14.0 - 14.0i)T + 343iT^{2} \) |
| 11 | \( 1 + (3.78 - 9.14i)T + (-941. - 941. i)T^{2} \) |
| 13 | \( 1 + (64.7 - 26.8i)T + (1.55e3 - 1.55e3i)T^{2} \) |
| 17 | \( 1 + 79.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (-94.7 + 39.2i)T + (4.85e3 - 4.85e3i)T^{2} \) |
| 23 | \( 1 + (-71.6 + 71.6i)T - 1.21e4iT^{2} \) |
| 29 | \( 1 + (53.0 + 128. i)T + (-1.72e4 + 1.72e4i)T^{2} \) |
| 31 | \( 1 + 267.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-205. - 85.1i)T + (3.58e4 + 3.58e4i)T^{2} \) |
| 41 | \( 1 + (-210. + 210. i)T - 6.89e4iT^{2} \) |
| 43 | \( 1 + (56.9 - 137. i)T + (-5.62e4 - 5.62e4i)T^{2} \) |
| 47 | \( 1 - 173. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + (188. - 455. i)T + (-1.05e5 - 1.05e5i)T^{2} \) |
| 59 | \( 1 + (627. + 260. i)T + (1.45e5 + 1.45e5i)T^{2} \) |
| 61 | \( 1 + (-66.5 - 160. i)T + (-1.60e5 + 1.60e5i)T^{2} \) |
| 67 | \( 1 + (-211. - 511. i)T + (-2.12e5 + 2.12e5i)T^{2} \) |
| 71 | \( 1 + (226. + 226. i)T + 3.57e5iT^{2} \) |
| 73 | \( 1 + (-802. + 802. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 - 552. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (137. - 57.1i)T + (4.04e5 - 4.04e5i)T^{2} \) |
| 89 | \( 1 + (579. + 579. i)T + 7.04e5iT^{2} \) |
| 97 | \( 1 + 912.T + 9.12e5T^{2} \) |
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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
−16.51347563823395052700647843365, −15.31771546796838425590220761905, −14.14898603103590793271817140012, −12.35470138456344207871630400927, −11.23568822204908037363415876516, −9.718544396568199919386055521259, −9.205420038696481369188440713029, −7.43728460658032912316520494440, −4.72311710534669232950365089377, −2.55570851254599094353503804800,
1.54753360798929950060122196433, 5.36192480552200443813078302459, 7.28673900547157943196433824553, 7.934729253887014372420054876225, 9.615374420088432675383048920325, 10.93793139880564774812438028678, 12.85245227488302967851508804872, 14.01495054107513901799918211149, 14.91108856000613523249250253449, 16.59171041699828319636981345426
