| L(s) = 1 | + (0.986 + 1.01i)2-s + (0.881 + 0.471i)3-s + (−0.0550 + 1.99i)4-s + (−0.677 + 0.825i)5-s + (0.391 + 1.35i)6-s + (1.41 + 0.942i)7-s + (−2.08 + 1.91i)8-s + (0.555 + 0.831i)9-s + (−1.50 + 0.127i)10-s + (−1.74 − 0.528i)11-s + (−0.991 + 1.73i)12-s + (2.10 − 1.72i)13-s + (0.435 + 2.35i)14-s + (−0.987 + 0.408i)15-s + (−3.99 − 0.220i)16-s + (−1.28 − 0.531i)17-s + ⋯ |
| L(s) = 1 | + (0.697 + 0.716i)2-s + (0.509 + 0.272i)3-s + (−0.0275 + 0.999i)4-s + (−0.303 + 0.369i)5-s + (0.159 + 0.554i)6-s + (0.532 + 0.356i)7-s + (−0.735 + 0.677i)8-s + (0.185 + 0.277i)9-s + (−0.476 + 0.0402i)10-s + (−0.525 − 0.159i)11-s + (−0.286 + 0.501i)12-s + (0.582 − 0.478i)13-s + (0.116 + 0.630i)14-s + (−0.254 + 0.105i)15-s + (−0.998 − 0.0550i)16-s + (−0.311 − 0.128i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.338 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.24500 + 1.77110i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24500 + 1.77110i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.986 - 1.01i)T \) |
| 3 | \( 1 + (-0.881 - 0.471i)T \) |
| good | 5 | \( 1 + (0.677 - 0.825i)T + (-0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (-1.41 - 0.942i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (1.74 + 0.528i)T + (9.14 + 6.11i)T^{2} \) |
| 13 | \( 1 + (-2.10 + 1.72i)T + (2.53 - 12.7i)T^{2} \) |
| 17 | \( 1 + (1.28 + 0.531i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.136 + 1.38i)T + (-18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (-3.76 - 0.749i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (1.26 + 4.16i)T + (-24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (-0.429 + 0.429i)T - 31iT^{2} \) |
| 37 | \( 1 + (-9.15 + 0.902i)T + (36.2 - 7.21i)T^{2} \) |
| 41 | \( 1 + (-0.0409 + 0.205i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-0.656 + 0.350i)T + (23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (1.66 - 4.01i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-2.99 + 9.86i)T + (-44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (1.91 + 1.57i)T + (11.5 + 57.8i)T^{2} \) |
| 61 | \( 1 + (-6.69 + 12.5i)T + (-33.8 - 50.7i)T^{2} \) |
| 67 | \( 1 + (5.87 - 10.9i)T + (-37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (1.22 - 1.83i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (-4.58 + 3.06i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (-2.27 - 5.49i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (7.02 + 0.691i)T + (81.4 + 16.1i)T^{2} \) |
| 89 | \( 1 + (14.5 - 2.89i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (-7.28 + 7.28i)T - 97iT^{2} \) |
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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
−11.48957226474368456594356960873, −11.04476330351685157171844998532, −9.585608222422790701792064957093, −8.526341491186007356861932140910, −7.896576152031804732350559159583, −6.95242469739416326902466659184, −5.72774436141476622855915888569, −4.79626196138047377357566560957, −3.61900774918983177860402969574, −2.58031704336840290658352333519,
1.28450904500149154822904803008, 2.69073783774904859692863747365, 4.01908839830142223610661758730, 4.82020850013376936145445942416, 6.12279593196783262957449467140, 7.28596403164716075860441585977, 8.405604168614777756615093075802, 9.255350084311444488115264290387, 10.40786226298361561353566098163, 11.13850017092324574044274160445
