| L(s) = 1 | + (0.599 − 0.800i)2-s + (−0.00859 − 0.0230i)3-s + (−0.281 − 0.959i)4-s + (1.78 − 1.34i)5-s + (−0.0235 − 0.00692i)6-s + (−0.339 + 1.56i)7-s + (−0.936 − 0.349i)8-s + (2.26 − 1.96i)9-s + (−0.00677 − 2.23i)10-s + (−3.99 − 0.574i)11-s + (−0.0196 + 0.0147i)12-s + (5.51 − 1.19i)13-s + (1.04 + 1.20i)14-s + (−0.0463 − 0.0295i)15-s + (−0.841 + 0.540i)16-s + (−3.06 + 1.67i)17-s + ⋯ |
| L(s) = 1 | + (0.423 − 0.566i)2-s + (−0.00496 − 0.0132i)3-s + (−0.140 − 0.479i)4-s + (0.798 − 0.601i)5-s + (−0.00963 − 0.00282i)6-s + (−0.128 + 0.589i)7-s + (−0.331 − 0.123i)8-s + (0.755 − 0.654i)9-s + (−0.00214 − 0.707i)10-s + (−1.20 − 0.173i)11-s + (−0.00568 + 0.00425i)12-s + (1.52 − 0.332i)13-s + (0.279 + 0.322i)14-s + (−0.0119 − 0.00763i)15-s + (−0.210 + 0.135i)16-s + (−0.743 + 0.406i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.370 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.370 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.36018 - 0.921615i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.36018 - 0.921615i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.599 + 0.800i)T \) |
| 5 | \( 1 + (-1.78 + 1.34i)T \) |
| 23 | \( 1 + (-4.66 + 1.13i)T \) |
| good | 3 | \( 1 + (0.00859 + 0.0230i)T + (-2.26 + 1.96i)T^{2} \) |
| 7 | \( 1 + (0.339 - 1.56i)T + (-6.36 - 2.90i)T^{2} \) |
| 11 | \( 1 + (3.99 + 0.574i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-5.51 + 1.19i)T + (11.8 - 5.40i)T^{2} \) |
| 17 | \( 1 + (3.06 - 1.67i)T + (9.19 - 14.3i)T^{2} \) |
| 19 | \( 1 + (3.86 - 1.13i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (2.58 - 8.80i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-1.61 - 3.52i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-0.746 - 0.0534i)T + (36.6 + 5.26i)T^{2} \) |
| 41 | \( 1 + (6.83 - 7.89i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (0.687 - 0.256i)T + (32.4 - 28.1i)T^{2} \) |
| 47 | \( 1 + (-0.892 - 0.892i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.75 + 1.03i)T + (48.2 + 22.0i)T^{2} \) |
| 59 | \( 1 + (3.63 - 5.65i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (4.49 - 2.05i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-12.1 - 9.09i)T + (18.8 + 64.2i)T^{2} \) |
| 71 | \( 1 + (1.14 + 7.98i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-2.12 + 3.89i)T + (-39.4 - 61.4i)T^{2} \) |
| 79 | \( 1 + (10.1 + 6.52i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-0.574 + 8.02i)T + (-82.1 - 11.8i)T^{2} \) |
| 89 | \( 1 + (-6.90 + 15.1i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-0.920 - 12.8i)T + (-96.0 + 13.8i)T^{2} \) |
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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
−12.33471063695811146837212930842, −10.91115619274480066717442656499, −10.33144758103197363283872112375, −9.088488207751326482454872543989, −8.472502773369177788060448278014, −6.59241710779560937101418119355, −5.71414352151648201327274375900, −4.61810381610646651822583706786, −3.11895185796634615532128472189, −1.53105696631186728570444836526,
2.29772170508595678365786932323, 3.94992662015572686249379739477, 5.17369798291130281491936263077, 6.36769290492293881693490650340, 7.16458105657939125899146088742, 8.238297879222349809612571020359, 9.528156372935549006590606526728, 10.60629925607878653584298200025, 11.15701128545620219960184564085, 12.95364644697425471029975490542
