| L(s) = 1 | + (−0.104 − 0.994i)2-s + (0.669 − 0.743i)3-s + (−0.978 + 0.207i)4-s + (1.06 − 0.474i)5-s + (−0.809 − 0.587i)6-s + (−1.26 − 2.32i)7-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (−0.583 − 1.01i)10-s + (−2.83 − 1.72i)11-s + (−0.5 + 0.866i)12-s + (0.534 − 0.388i)13-s + (−2.17 + 1.50i)14-s + (0.360 − 1.11i)15-s + (0.913 − 0.406i)16-s + (0.0238 − 0.226i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 − 0.703i)2-s + (0.386 − 0.429i)3-s + (−0.489 + 0.103i)4-s + (0.476 − 0.212i)5-s + (−0.330 − 0.239i)6-s + (−0.478 − 0.878i)7-s + (0.109 + 0.336i)8-s + (−0.0348 − 0.331i)9-s + (−0.184 − 0.319i)10-s + (−0.853 − 0.521i)11-s + (−0.144 + 0.249i)12-s + (0.148 − 0.107i)13-s + (−0.582 + 0.401i)14-s + (0.0931 − 0.286i)15-s + (0.228 − 0.101i)16-s + (0.00577 − 0.0549i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 + 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.814 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.395944 - 1.23999i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.395944 - 1.23999i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 + 0.994i)T \) |
| 3 | \( 1 + (-0.669 + 0.743i)T \) |
| 7 | \( 1 + (1.26 + 2.32i)T \) |
| 11 | \( 1 + (2.83 + 1.72i)T \) |
| good | 5 | \( 1 + (-1.06 + 0.474i)T + (3.34 - 3.71i)T^{2} \) |
| 13 | \( 1 + (-0.534 + 0.388i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.0238 + 0.226i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (0.381 + 0.0811i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-4.34 + 7.53i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.834 + 2.56i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.169 - 0.0755i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-2.59 - 2.88i)T + (-3.86 + 36.7i)T^{2} \) |
| 41 | \( 1 + (0.155 + 0.477i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 4.81T + 43T^{2} \) |
| 47 | \( 1 + (-1.13 - 0.240i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-8.54 - 3.80i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (-10.2 + 2.18i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (8.69 - 3.87i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-2.15 - 3.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.35 - 3.89i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-14.3 + 3.04i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (0.0118 + 0.112i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (8.46 + 6.15i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-6.30 + 10.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.08 + 4.41i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.56982426216304409220995434920, −9.969633224589290490501073262226, −8.943333424529087401719474126500, −8.128579040709294607812851450826, −7.09517619295205445311308334972, −5.99608283833355943759703407321, −4.72818119783240017907682847753, −3.45997920973020173605459388568, −2.41224301741095777327475672656, −0.796242385412229497495456053352,
2.25838207041528357834292710809, 3.51487859799935413405125386151, 4.99534410551789342002674895911, 5.71178744488395928723962917150, 6.78883011809033710967739790312, 7.80944835078323138159173845346, 8.759440123418135950724519684768, 9.561771932645062845737986706476, 10.14812760874502308922726550683, 11.28293709999491626138072159122
