| L(s) = 1 | + (−25.8 + 18.8i)2-s + (228. − 701. i)3-s + (316. − 973. i)4-s + (−4.89e3 + 4.98e3i)5-s + (7.29e3 + 2.24e4i)6-s + 8.24e3·7-s + (1.01e4 + 3.11e4i)8-s + (−2.97e5 − 2.15e5i)9-s + (3.31e4 − 2.21e5i)10-s + (2.27e5 − 1.65e5i)11-s + (−6.11e5 − 4.44e5i)12-s + (2.08e6 + 1.51e6i)13-s + (−2.13e5 + 1.55e5i)14-s + (2.37e6 + 4.57e6i)15-s + (−8.48e5 − 6.16e5i)16-s + (2.85e6 + 8.77e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.541 − 1.66i)3-s + (0.154 − 0.475i)4-s + (−0.701 + 0.712i)5-s + (0.383 + 1.17i)6-s + 0.185·7-s + (0.109 + 0.336i)8-s + (−1.67 − 1.21i)9-s + (0.104 − 0.699i)10-s + (0.425 − 0.309i)11-s + (−0.709 − 0.515i)12-s + (1.55 + 1.13i)13-s + (−0.106 + 0.0770i)14-s + (0.808 + 1.55i)15-s + (−0.202 − 0.146i)16-s + (0.487 + 1.49i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.60079 - 0.676895i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60079 - 0.676895i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (25.8 - 18.8i)T \) |
| 5 | \( 1 + (4.89e3 - 4.98e3i)T \) |
| good | 3 | \( 1 + (-228. + 701. i)T + (-1.43e5 - 1.04e5i)T^{2} \) |
| 7 | \( 1 - 8.24e3T + 1.97e9T^{2} \) |
| 11 | \( 1 + (-2.27e5 + 1.65e5i)T + (8.81e10 - 2.71e11i)T^{2} \) |
| 13 | \( 1 + (-2.08e6 - 1.51e6i)T + (5.53e11 + 1.70e12i)T^{2} \) |
| 17 | \( 1 + (-2.85e6 - 8.77e6i)T + (-2.77e13 + 2.01e13i)T^{2} \) |
| 19 | \( 1 + (1.93e6 + 5.94e6i)T + (-9.42e13 + 6.84e13i)T^{2} \) |
| 23 | \( 1 + (-2.51e7 + 1.82e7i)T + (2.94e14 - 9.06e14i)T^{2} \) |
| 29 | \( 1 + (3.49e6 - 1.07e7i)T + (-9.87e15 - 7.17e15i)T^{2} \) |
| 31 | \( 1 + (-1.96e7 - 6.06e7i)T + (-2.05e16 + 1.49e16i)T^{2} \) |
| 37 | \( 1 + (-4.96e7 - 3.60e7i)T + (5.49e16 + 1.69e17i)T^{2} \) |
| 41 | \( 1 + (6.98e7 + 5.07e7i)T + (1.70e17 + 5.23e17i)T^{2} \) |
| 43 | \( 1 + 1.23e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-3.73e8 + 1.15e9i)T + (-2.00e18 - 1.45e18i)T^{2} \) |
| 53 | \( 1 + (-1.21e9 + 3.74e9i)T + (-7.49e18 - 5.44e18i)T^{2} \) |
| 59 | \( 1 + (-6.96e9 - 5.06e9i)T + (9.31e18 + 2.86e19i)T^{2} \) |
| 61 | \( 1 + (-5.70e9 + 4.14e9i)T + (1.34e19 - 4.13e19i)T^{2} \) |
| 67 | \( 1 + (4.82e9 + 1.48e10i)T + (-9.88e19 + 7.17e19i)T^{2} \) |
| 71 | \( 1 + (2.17e9 - 6.69e9i)T + (-1.86e20 - 1.35e20i)T^{2} \) |
| 73 | \( 1 + (-1.36e10 + 9.92e9i)T + (9.69e19 - 2.98e20i)T^{2} \) |
| 79 | \( 1 + (-4.59e9 + 1.41e10i)T + (-6.05e20 - 4.39e20i)T^{2} \) |
| 83 | \( 1 + (-2.18e10 - 6.73e10i)T + (-1.04e21 + 7.56e20i)T^{2} \) |
| 89 | \( 1 + (-7.69e10 + 5.59e10i)T + (8.57e20 - 2.63e21i)T^{2} \) |
| 97 | \( 1 + (-2.89e10 + 8.92e10i)T + (-5.78e21 - 4.20e21i)T^{2} \) |
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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
−13.16206218637001651549212342978, −11.81611101857799642641160037134, −10.88478758718471868021928679361, −8.747195144482823791326069046915, −8.162354763240551638790566569757, −6.85621010310949647825333181504, −6.32775719317711004595317387983, −3.56436456672556692040090453757, −1.91842180880914442653625353400, −0.830241126949970532133028798292,
0.908732074050500449103450150343, 3.11889673735054331058532630911, 4.01499415649723678573855318917, 5.28300987200633691294721358735, 7.85700856984556327910117928077, 8.796298765274575117752129638329, 9.639612245901811738560920850118, 10.81167383468153955177216862026, 11.70637636619247008592715102415, 13.29630575792615082072244967559
