| L(s) = 1 | + (−1.33 − 0.462i)2-s + (0.536 − 1.64i)3-s + (1.57 + 1.23i)4-s + (−3.31 + 1.51i)5-s + (−1.47 + 1.95i)6-s + (0.404 + 0.423i)7-s + (−1.52 − 2.37i)8-s + (−2.42 − 1.76i)9-s + (5.12 − 0.489i)10-s + (3.94 + 0.376i)11-s + (2.87 − 1.92i)12-s + (1.58 + 0.305i)13-s + (−0.343 − 0.753i)14-s + (0.713 + 6.26i)15-s + (0.944 + 3.88i)16-s + (1.13 − 2.19i)17-s + ⋯ |
| L(s) = 1 | + (−0.945 − 0.327i)2-s + (0.309 − 0.950i)3-s + (0.786 + 0.618i)4-s + (−1.48 + 0.676i)5-s + (−0.603 + 0.797i)6-s + (0.152 + 0.160i)7-s + (−0.540 − 0.841i)8-s + (−0.808 − 0.589i)9-s + (1.62 − 0.154i)10-s + (1.19 + 0.113i)11-s + (0.831 − 0.556i)12-s + (0.439 + 0.0846i)13-s + (−0.0919 − 0.201i)14-s + (0.184 + 1.61i)15-s + (0.236 + 0.971i)16-s + (0.274 − 0.532i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.345043 - 0.610097i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.345043 - 0.610097i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.33 + 0.462i)T \) |
| 3 | \( 1 + (-0.536 + 1.64i)T \) |
| 67 | \( 1 + (-6.83 + 4.49i)T \) |
| good | 5 | \( 1 + (3.31 - 1.51i)T + (3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (-0.404 - 0.423i)T + (-0.333 + 6.99i)T^{2} \) |
| 11 | \( 1 + (-3.94 - 0.376i)T + (10.8 + 2.08i)T^{2} \) |
| 13 | \( 1 + (-1.58 - 0.305i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (-1.13 + 2.19i)T + (-9.86 - 13.8i)T^{2} \) |
| 19 | \( 1 + (-1.03 + 1.08i)T + (-0.904 - 18.9i)T^{2} \) |
| 23 | \( 1 + (0.773 - 0.309i)T + (16.6 - 15.8i)T^{2} \) |
| 29 | \( 1 + (4.75 + 2.74i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.31 + 6.80i)T + (-28.7 + 11.5i)T^{2} \) |
| 37 | \( 1 + (0.495 + 0.858i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.27 + 0.251i)T + (40.8 - 3.89i)T^{2} \) |
| 43 | \( 1 + (3.38 + 5.25i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (4.91 + 3.86i)T + (11.0 + 45.6i)T^{2} \) |
| 53 | \( 1 + (-4.01 + 6.24i)T + (-22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-4.22 - 4.87i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (8.94 - 0.853i)T + (59.8 - 11.5i)T^{2} \) |
| 71 | \( 1 + (7.15 - 3.68i)T + (41.1 - 57.8i)T^{2} \) |
| 73 | \( 1 + (-7.55 + 0.721i)T + (71.6 - 13.8i)T^{2} \) |
| 79 | \( 1 + (6.27 - 2.17i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (-6.78 + 9.52i)T + (-27.1 - 78.4i)T^{2} \) |
| 89 | \( 1 + (1.42 - 0.205i)T + (85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (9.18 + 15.9i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.845570487262818251330742310468, −8.919081444917626233640158901668, −8.247296823794060864831531660330, −7.40289042145487419546347728488, −7.02276463952143002951834186076, −6.03083765337416519014995424954, −3.95952814091313567071679196917, −3.28767488116394832516423100592, −2.01655167339895272901112496609, −0.51762840748967914153174641689,
1.28152604244947796316343780430, 3.30573188561787815956523454973, 4.10263627677063787166753486012, 5.13502908284083294808132624475, 6.30122988542058575937783885661, 7.52472599223593118864586408347, 8.150463994066368951649074884937, 8.863146821417208133832671222308, 9.397661653041644932733307481287, 10.51488844118715716013684620780
