| L(s) = 1 | + (82.0 + 142. i)3-s + (1.67e3 − 2.89e3i)5-s + (2.63e4 − 3.57e4i)7-s + (7.51e4 − 1.30e5i)9-s + (1.78e5 + 3.09e5i)11-s + 2.06e6·13-s + 5.47e5·15-s + (1.53e6 + 2.65e6i)17-s + (−6.51e6 + 1.12e7i)19-s + (7.24e6 + 8.12e5i)21-s + (2.24e7 − 3.88e7i)23-s + (1.88e7 + 3.26e7i)25-s + 5.37e7·27-s − 1.16e8·29-s + (−1.09e8 − 1.90e8i)31-s + ⋯ |
| L(s) = 1 | + (0.194 + 0.337i)3-s + (0.239 − 0.414i)5-s + (0.593 − 0.804i)7-s + (0.424 − 0.734i)9-s + (0.334 + 0.578i)11-s + 1.54·13-s + 0.186·15-s + (0.262 + 0.454i)17-s + (−0.603 + 1.04i)19-s + (0.387 + 0.0434i)21-s + (0.727 − 1.26i)23-s + (0.385 + 0.668i)25-s + 0.720·27-s − 1.05·29-s + (−0.689 − 1.19i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(3.215198116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.215198116\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.63e4 + 3.57e4i)T \) |
| good | 3 | \( 1 + (-82.0 - 142. i)T + (-8.85e4 + 1.53e5i)T^{2} \) |
| 5 | \( 1 + (-1.67e3 + 2.89e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-1.78e5 - 3.09e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 - 2.06e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-1.53e6 - 2.65e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (6.51e6 - 1.12e7i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-2.24e7 + 3.88e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 + 1.16e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + (1.09e8 + 1.90e8i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (3.99e7 - 6.91e7i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 - 9.01e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 5.79e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-5.34e8 + 9.24e8i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-2.00e8 - 3.46e8i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-2.26e9 - 3.92e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (8.45e8 - 1.46e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-4.36e9 - 7.56e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 - 5.61e8T + 2.31e20T^{2} \) |
| 73 | \( 1 + (-4.25e9 - 7.37e9i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-1.02e10 + 1.77e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 - 2.59e8T + 1.28e21T^{2} \) |
| 89 | \( 1 + (4.21e10 - 7.30e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 - 1.58e11T + 7.15e21T^{2} \) |
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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
−11.19164564705198084202573867414, −10.33403289863577420180997048282, −9.224009063532954856864772326924, −8.305814189040924839995534825496, −7.01024962510749752444483452949, −5.83101720352544458843244881714, −4.31076218397226697894135511675, −3.69254430905824369956231640365, −1.70772686429050537408920980497, −0.849967165314422981352348961525,
1.10643144277136828645747489055, 2.14099590578259229634108526638, 3.37643922329164145532094728696, 4.96187205820018263879015157947, 6.07167027972388308218578008213, 7.24246930042899372550732260928, 8.423475719572849041162160164955, 9.210984007469918108362042282812, 10.85633621192285756078868945972, 11.28236792264735987320220213612
