| L(s) = 1 | + (0.358 − 0.485i)2-s + (−0.0479 + 0.00906i)3-s + (0.482 + 1.56i)4-s + (−2.11 + 0.726i)5-s + (−0.0127 + 0.0265i)6-s + (−0.483 + 2.60i)7-s + (2.07 + 0.724i)8-s + (−2.79 + 1.09i)9-s + (−0.404 + 1.28i)10-s + (−0.343 + 0.874i)11-s + (−0.0372 − 0.0705i)12-s + (5.89 + 0.663i)13-s + (1.08 + 1.16i)14-s + (0.0947 − 0.0540i)15-s + (−1.61 + 1.09i)16-s + (−4.55 − 0.170i)17-s + ⋯ |
| L(s) = 1 | + (0.253 − 0.343i)2-s + (−0.0276 + 0.00523i)3-s + (0.241 + 0.781i)4-s + (−0.945 + 0.325i)5-s + (−0.00521 + 0.0108i)6-s + (−0.182 + 0.983i)7-s + (0.731 + 0.256i)8-s + (−0.930 + 0.365i)9-s + (−0.128 + 0.406i)10-s + (−0.103 + 0.263i)11-s + (−0.0107 − 0.0203i)12-s + (1.63 + 0.184i)13-s + (0.291 + 0.311i)14-s + (0.0244 − 0.0139i)15-s + (−0.402 + 0.274i)16-s + (−1.10 − 0.0413i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.307 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.952177 + 0.693102i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.952177 + 0.693102i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.11 - 0.726i)T \) |
| 7 | \( 1 + (0.483 - 2.60i)T \) |
| good | 2 | \( 1 + (-0.358 + 0.485i)T + (-0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (0.0479 - 0.00906i)T + (2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (0.343 - 0.874i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-5.89 - 0.663i)T + (12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (4.55 + 0.170i)T + (16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (-4.20 + 7.28i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.248 - 6.64i)T + (-22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (-1.80 + 0.411i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-3.48 + 2.01i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.67 + 1.93i)T + (20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (0.726 + 1.50i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-1.02 - 2.91i)T + (-33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (3.22 + 2.37i)T + (13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (-5.37 - 2.83i)T + (29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (0.410 + 5.47i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (2.74 - 8.90i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (6.43 + 1.72i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-2.32 + 10.1i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-3.05 + 2.25i)T + (21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (-0.820 - 0.473i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.767 - 6.80i)T + (-80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (-2.08 - 5.30i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-8.43 - 8.43i)T + 97iT^{2} \) |
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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
−11.98004090581613642063567315923, −11.36975150829692935402737945965, −11.02191444989966099486563841611, −9.094138444388998337876974293566, −8.432186013826479966441325681793, −7.42863998318235525018713162159, −6.27110041044264661103667868094, −4.81036264594687435218837587258, −3.47594745693585836216519901903, −2.56042974943876523929699589190,
0.930846990992109996118258057513, 3.45227723787144477714023335942, 4.52189488326407591328280673030, 5.91547599390711809498976640133, 6.68214447507649776367733085820, 7.967874969965423367539922137755, 8.773643264518553360857997874446, 10.23469599468157690163572330064, 10.97979814252011416607593664808, 11.71085455672071379065270952410
