| L(s) = 1 | + (−0.930 − 0.365i)2-s + (−1.53 − 0.802i)3-s + (0.733 + 0.680i)4-s + (2.55 + 1.74i)5-s + (1.13 + 1.30i)6-s + (−2.00 − 1.72i)7-s + (−0.433 − 0.900i)8-s + (1.71 + 2.46i)9-s + (−1.74 − 2.55i)10-s + (0.534 + 3.54i)11-s + (−0.578 − 1.63i)12-s + (−2.98 + 2.37i)13-s + (1.23 + 2.33i)14-s + (−2.52 − 4.73i)15-s + (0.0747 + 0.997i)16-s + (5.76 + 1.77i)17-s + ⋯ |
| L(s) = 1 | + (−0.658 − 0.258i)2-s + (−0.886 − 0.463i)3-s + (0.366 + 0.340i)4-s + (1.14 + 0.780i)5-s + (0.463 + 0.534i)6-s + (−0.758 − 0.651i)7-s + (−0.153 − 0.318i)8-s + (0.570 + 0.821i)9-s + (−0.551 − 0.809i)10-s + (0.161 + 1.06i)11-s + (−0.167 − 0.471i)12-s + (−0.827 + 0.659i)13-s + (0.330 + 0.624i)14-s + (−0.652 − 1.22i)15-s + (0.0186 + 0.249i)16-s + (1.39 + 0.431i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.797477 + 0.129108i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.797477 + 0.129108i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.930 + 0.365i)T \) |
| 3 | \( 1 + (1.53 + 0.802i)T \) |
| 7 | \( 1 + (2.00 + 1.72i)T \) |
| good | 5 | \( 1 + (-2.55 - 1.74i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (-0.534 - 3.54i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (2.98 - 2.37i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-5.76 - 1.77i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-5.02 + 2.90i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.630 - 2.04i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-5.09 + 1.16i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-6.99 - 4.03i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.31 - 3.07i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (-1.73 + 0.833i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (4.97 + 2.39i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (1.88 - 4.79i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (2.48 - 2.67i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (3.09 - 2.10i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (7.51 + 8.09i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (1.78 - 3.09i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7.11 + 1.62i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-13.1 + 5.17i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (0.673 + 1.16i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.57 + 4.48i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (0.738 + 0.111i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + 15.4iT - 97T^{2} \) |
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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
−11.87747856587144022976834905484, −10.59642366130042698691465408767, −9.970633398076152855468412373552, −9.568298347795332028893253131458, −7.62477753828152545242775343165, −6.92068287845892565724925705367, −6.24274019025887443971269814017, −4.86830958282936873738719338509, −2.93122307502518988376408300801, −1.47858042022197093288500669068,
0.938867817285250233962596588138, 3.06259354329411228787976716789, 5.20776805277968242543174721645, 5.66216890923283464077150812331, 6.53059967431716912742018393164, 8.045804437130530107246808537474, 9.236292672348707901973206304664, 9.800307323264902715895443649140, 10.37981857786837389457678879097, 11.84801906279050742848170934514
