| L(s) = 1 | + (0.210 + 1.39i)2-s + (0.453 + 0.891i)3-s + (−1.91 + 0.588i)4-s + (−0.590 + 2.15i)5-s + (−1.15 + 0.822i)6-s + (1.77 + 1.77i)7-s + (−1.22 − 2.54i)8-s + (−0.587 + 0.809i)9-s + (−3.14 − 0.371i)10-s + (0.582 + 0.801i)11-s + (−1.39 − 1.43i)12-s + (−0.710 − 4.48i)13-s + (−2.10 + 2.85i)14-s + (−2.18 + 0.453i)15-s + (3.30 − 2.24i)16-s + (−3.22 − 1.64i)17-s + ⋯ |
| L(s) = 1 | + (0.148 + 0.988i)2-s + (0.262 + 0.514i)3-s + (−0.955 + 0.294i)4-s + (−0.263 + 0.964i)5-s + (−0.469 + 0.335i)6-s + (0.670 + 0.670i)7-s + (−0.432 − 0.901i)8-s + (−0.195 + 0.269i)9-s + (−0.993 − 0.117i)10-s + (0.175 + 0.241i)11-s + (−0.401 − 0.414i)12-s + (−0.197 − 1.24i)13-s + (−0.563 + 0.763i)14-s + (−0.565 + 0.117i)15-s + (0.827 − 0.562i)16-s + (−0.781 − 0.398i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.170947 + 1.27246i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.170947 + 1.27246i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.210 - 1.39i)T \) |
| 3 | \( 1 + (-0.453 - 0.891i)T \) |
| 5 | \( 1 + (0.590 - 2.15i)T \) |
| good | 7 | \( 1 + (-1.77 - 1.77i)T + 7iT^{2} \) |
| 11 | \( 1 + (-0.582 - 0.801i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.710 + 4.48i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (3.22 + 1.64i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-0.911 - 2.80i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (1.25 - 7.89i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-0.616 - 0.200i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-6.41 + 2.08i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.16 + 0.658i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-9.50 - 6.90i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-1.10 + 1.10i)T - 43iT^{2} \) |
| 47 | \( 1 + (-8.43 + 4.30i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (5.38 - 2.74i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (5.91 + 4.29i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.00 + 0.727i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (2.66 - 5.23i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-11.9 - 3.87i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (8.37 + 1.32i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-2.78 + 8.57i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.13 - 1.59i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-3.29 - 4.53i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-3.83 - 7.53i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−12.15340812058760591468651665012, −11.22715870204315253056964615370, −10.08888766717280661259955103808, −9.267884938101297712031517904666, −8.046002741456998413939188788224, −7.56977226581969375906793431899, −6.19668666151638660589876840548, −5.28375749152789150086233402475, −4.07672814966474538687407933152, −2.81989908046061473216800287659,
0.957221695591761389582457915259, 2.32240469946453144082291749001, 4.18327652780622124492930488721, 4.66611367626050036696895847884, 6.30814538127669252617902542668, 7.73152473100678063995417532364, 8.697476412973880580148056735591, 9.271708329318198027167681417661, 10.63731497325799911771166057953, 11.43508250519253153767500471216
