| L(s) = 1 | + (−0.744 + 0.593i)2-s + (−1.04 + 1.38i)3-s + (−0.243 + 1.06i)4-s + (0.397 + 0.271i)5-s + (−0.0452 − 1.64i)6-s + (−2.38 − 1.15i)7-s + (−1.27 − 2.65i)8-s + (−0.827 − 2.88i)9-s + (−0.457 + 0.0342i)10-s + (−0.227 − 1.51i)11-s + (−1.22 − 1.44i)12-s + (0.172 + 1.14i)13-s + (2.45 − 0.553i)14-s + (−0.789 + 0.267i)15-s + (0.557 + 0.268i)16-s + (1.74 − 1.61i)17-s + ⋯ |
| L(s) = 1 | + (−0.526 + 0.419i)2-s + (−0.601 + 0.798i)3-s + (−0.121 + 0.532i)4-s + (0.177 + 0.121i)5-s + (−0.0184 − 0.673i)6-s + (−0.899 − 0.436i)7-s + (−0.451 − 0.938i)8-s + (−0.275 − 0.961i)9-s + (−0.144 + 0.0108i)10-s + (−0.0686 − 0.455i)11-s + (−0.352 − 0.417i)12-s + (0.0478 + 0.317i)13-s + (0.656 − 0.148i)14-s + (−0.203 + 0.0690i)15-s + (0.139 + 0.0670i)16-s + (0.422 − 0.392i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.480 + 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.480 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.207186 - 0.122663i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.207186 - 0.122663i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.04 - 1.38i)T \) |
| 7 | \( 1 + (2.38 + 1.15i)T \) |
| good | 2 | \( 1 + (0.744 - 0.593i)T + (0.445 - 1.94i)T^{2} \) |
| 5 | \( 1 + (-0.397 - 0.271i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (0.227 + 1.51i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (-0.172 - 1.14i)T + (-12.4 + 3.83i)T^{2} \) |
| 17 | \( 1 + (-1.74 + 1.61i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (5.11 + 2.95i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.309 - 1.00i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-6.27 - 6.75i)T + (-2.16 + 28.9i)T^{2} \) |
| 31 | \( 1 - 7.44iT - 31T^{2} \) |
| 37 | \( 1 + (8.67 + 2.67i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-0.620 + 8.28i)T + (-40.5 - 6.11i)T^{2} \) |
| 43 | \( 1 + (0.683 + 9.11i)T + (-42.5 + 6.40i)T^{2} \) |
| 47 | \( 1 + (6.24 + 7.83i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (3.06 + 9.94i)T + (-43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-3.49 - 1.68i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (-1.61 + 0.369i)T + (54.9 - 26.4i)T^{2} \) |
| 67 | \( 1 + 9.68T + 67T^{2} \) |
| 71 | \( 1 + (7.81 + 1.78i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (0.768 - 5.10i)T + (-69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + (-10.1 - 1.53i)T + (79.3 + 24.4i)T^{2} \) |
| 89 | \( 1 + (-2.24 - 5.70i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (3.50 - 2.02i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.55527287438754867711592156043, −10.18816483418759277330926040515, −9.015620645504380019604965762650, −8.584634809051675366101321805901, −6.93266273100044718751985988224, −6.62253437747114834312301464588, −5.26581021969027093944580941560, −3.98553899487023209850691739239, −3.13339330076074617546152526604, −0.19565040126743086175462305250,
1.50329777204058291811985817027, 2.71111778399566537289359850381, 4.68193636425432356486778925166, 6.01734814196781876528818416531, 6.21490187579653060488009155192, 7.73735318295464681177408537384, 8.601819229596234208898282468026, 9.753497009274508393790982266681, 10.24877345269838832204176850261, 11.25428734951880814817092588072
