| L(s) = 1 | + (−0.165 + 0.872i)2-s + (0.0256 − 0.684i)3-s + (1.12 + 0.442i)4-s + (−2.22 + 0.192i)5-s + (0.593 + 0.135i)6-s + (−1.59 + 2.11i)7-s + (−1.51 + 2.41i)8-s + (2.52 + 0.189i)9-s + (0.199 − 1.97i)10-s + (0.385 + 5.14i)11-s + (0.331 − 0.760i)12-s + (1.06 − 0.373i)13-s + (−1.57 − 1.74i)14-s + (0.0748 + 1.53i)15-s + (−0.0817 − 0.0758i)16-s + (2.08 + 2.82i)17-s + ⋯ |
| L(s) = 1 | + (−0.116 + 0.617i)2-s + (0.0147 − 0.395i)3-s + (0.563 + 0.221i)4-s + (−0.996 + 0.0861i)5-s + (0.242 + 0.0552i)6-s + (−0.603 + 0.797i)7-s + (−0.536 + 0.853i)8-s + (0.841 + 0.0630i)9-s + (0.0631 − 0.624i)10-s + (0.116 + 1.54i)11-s + (0.0958 − 0.219i)12-s + (0.295 − 0.103i)13-s + (−0.421 − 0.465i)14-s + (0.0193 + 0.395i)15-s + (−0.0204 − 0.0189i)16-s + (0.506 + 0.685i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0969 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0969 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.754806 + 0.831928i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.754806 + 0.831928i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.22 - 0.192i)T \) |
| 7 | \( 1 + (1.59 - 2.11i)T \) |
| good | 2 | \( 1 + (0.165 - 0.872i)T + (-1.86 - 0.730i)T^{2} \) |
| 3 | \( 1 + (-0.0256 + 0.684i)T + (-2.99 - 0.224i)T^{2} \) |
| 11 | \( 1 + (-0.385 - 5.14i)T + (-10.8 + 1.63i)T^{2} \) |
| 13 | \( 1 + (-1.06 + 0.373i)T + (10.1 - 8.10i)T^{2} \) |
| 17 | \( 1 + (-2.08 - 2.82i)T + (-5.01 + 16.2i)T^{2} \) |
| 19 | \( 1 + (-3.18 + 5.52i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.40 + 1.77i)T + (6.77 + 21.9i)T^{2} \) |
| 29 | \( 1 + (1.81 + 1.44i)T + (6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (5.59 - 3.23i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.87 + 2.99i)T + (25.1 + 27.1i)T^{2} \) |
| 41 | \( 1 + (-2.71 + 0.619i)T + (36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-3.91 + 2.45i)T + (18.6 - 38.7i)T^{2} \) |
| 47 | \( 1 + (0.681 + 0.128i)T + (43.7 + 17.1i)T^{2} \) |
| 53 | \( 1 + (-10.8 + 4.74i)T + (36.0 - 38.8i)T^{2} \) |
| 59 | \( 1 + (1.39 - 0.428i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (-9.85 + 3.86i)T + (44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-10.3 - 2.78i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (1.16 + 1.46i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (3.23 - 0.611i)T + (67.9 - 26.6i)T^{2} \) |
| 79 | \( 1 + (-12.1 - 7.03i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.52 - 4.36i)T + (-64.8 - 51.7i)T^{2} \) |
| 89 | \( 1 + (-0.486 + 6.49i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + (-1.18 - 1.18i)T + 97iT^{2} \) |
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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
−12.38388750856447158906234634251, −11.67194628654827599055813230580, −10.46795588768974341282171390892, −9.251772312006468826024057869045, −8.149036984770559165046084652280, −7.17438197603427860886419269020, −6.79733453298481211733474726209, −5.31964991840168049612981842892, −3.77932317521215294539187543007, −2.23212561451825057791947402052,
1.00068175229509849924428002626, 3.39877401784294338196124178441, 3.81635300570944824015847620714, 5.69435766139278118883093487612, 6.95964423749405611196685161128, 7.79817645310457593424306940796, 9.228889591638612711497498782575, 10.14885512509537140860191816537, 10.89047475738731369012856261564, 11.67161928813710923413134933656
