L(s) = 1 | + (2.18 − 1.52i)2-s + (1.73 − 4.77i)4-s + (−1.64 + 1.51i)5-s + (1.30 − 2.78i)7-s + (−2.12 − 7.92i)8-s + (−1.28 + 5.81i)10-s + (0.426 − 0.508i)11-s + (−0.269 + 0.384i)13-s + (−1.42 − 8.06i)14-s + (−8.94 − 7.50i)16-s + (−1.20 + 4.50i)17-s + (4.80 + 2.77i)19-s + (4.34 + 10.5i)20-s + (0.154 − 1.76i)22-s + (0.0602 − 0.0280i)23-s + ⋯ |
L(s) = 1 | + (1.54 − 1.07i)2-s + (0.869 − 2.38i)4-s + (−0.737 + 0.675i)5-s + (0.491 − 1.05i)7-s + (−0.750 − 2.80i)8-s + (−0.407 + 1.83i)10-s + (0.128 − 0.153i)11-s + (−0.0746 + 0.106i)13-s + (−0.380 − 2.15i)14-s + (−2.23 − 1.87i)16-s + (−0.292 + 1.09i)17-s + (1.10 + 0.637i)19-s + (0.972 + 2.34i)20-s + (0.0328 − 0.375i)22-s + (0.0125 − 0.00585i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.66186 - 2.34605i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.66186 - 2.34605i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.64 - 1.51i)T \) |
good | 2 | \( 1 + (-2.18 + 1.52i)T + (0.684 - 1.87i)T^{2} \) |
| 7 | \( 1 + (-1.30 + 2.78i)T + (-4.49 - 5.36i)T^{2} \) |
| 11 | \( 1 + (-0.426 + 0.508i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.269 - 0.384i)T + (-4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (1.20 - 4.50i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-4.80 - 2.77i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0602 + 0.0280i)T + (14.7 - 17.6i)T^{2} \) |
| 29 | \( 1 + (0.434 - 2.46i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.76 - 0.642i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (2.44 + 0.656i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (1.62 - 0.286i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.732 - 8.36i)T + (-42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-3.71 - 1.73i)T + (30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (7.52 - 7.52i)T - 53iT^{2} \) |
| 59 | \( 1 + (-3.49 + 2.93i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (5.84 - 2.12i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (4.48 + 3.13i)T + (22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (-5.33 + 3.07i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-13.6 + 3.64i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (5.34 + 0.941i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (3.25 + 4.64i)T + (-28.3 + 77.9i)T^{2} \) |
| 89 | \( 1 + (-3.08 + 5.33i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (13.5 - 1.18i)T + (95.5 - 16.8i)T^{2} \) |
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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
−10.94651734947672233096711297708, −10.74738226297253276426358648490, −9.713475800888984100365190774942, −7.979216235079794447420594496673, −6.94723858858363522877646588867, −5.94977980629238301174746861706, −4.65283950859573844231999708278, −3.90154662152656348097049963388, −3.06092208227692981404136201494, −1.43163658792941009953552947446,
2.64690700998015598573477744532, 3.89789805854824542631945508793, 5.05767292715144592318576284610, 5.34791415791044157847094926186, 6.74414603358114757987878833911, 7.59332599156702575667137336630, 8.417627947138894565559342500776, 9.295116873170174149617365767046, 11.34548548163029308243137493582, 11.86996905399605425964482214695