| L(s) = 1 | + (2.70 − 1.30i)3-s + (0.364 − 1.59i)5-s + (−0.0422 + 0.0203i)7-s + (5.61 − 7.03i)9-s + (−5.76 − 7.22i)11-s + (−41.3 − 51.8i)13-s + (−1.09 − 4.79i)15-s + 49.0·17-s + (−33.2 − 16.0i)19-s + (−0.0876 + 0.109i)21-s + (−30.0 − 131. i)23-s + (110. + 53.0i)25-s + (6.00 − 26.3i)27-s + (−133. − 81.8i)29-s + (5.86 − 25.7i)31-s + ⋯ |
| L(s) = 1 | + (0.520 − 0.250i)3-s + (0.0326 − 0.142i)5-s + (−0.00227 + 0.00109i)7-s + (0.207 − 0.260i)9-s + (−0.157 − 0.198i)11-s + (−0.882 − 1.10i)13-s + (−0.0188 − 0.0825i)15-s + 0.700·17-s + (−0.401 − 0.193i)19-s + (−0.000910 + 0.00114i)21-s + (−0.272 − 1.19i)23-s + (0.881 + 0.424i)25-s + (0.0428 − 0.187i)27-s + (−0.851 − 0.524i)29-s + (0.0340 − 0.148i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.672047158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.672047158\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.70 + 1.30i)T \) |
| 29 | \( 1 + (133. + 81.8i)T \) |
| good | 5 | \( 1 + (-0.364 + 1.59i)T + (-112. - 54.2i)T^{2} \) |
| 7 | \( 1 + (0.0422 - 0.0203i)T + (213. - 268. i)T^{2} \) |
| 11 | \( 1 + (5.76 + 7.22i)T + (-296. + 1.29e3i)T^{2} \) |
| 13 | \( 1 + (41.3 + 51.8i)T + (-488. + 2.14e3i)T^{2} \) |
| 17 | \( 1 - 49.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + (33.2 + 16.0i)T + (4.27e3 + 5.36e3i)T^{2} \) |
| 23 | \( 1 + (30.0 + 131. i)T + (-1.09e4 + 5.27e3i)T^{2} \) |
| 31 | \( 1 + (-5.86 + 25.7i)T + (-2.68e4 - 1.29e4i)T^{2} \) |
| 37 | \( 1 + (-178. + 224. i)T + (-1.12e4 - 4.93e4i)T^{2} \) |
| 41 | \( 1 + 374.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (49.6 + 217. i)T + (-7.16e4 + 3.44e4i)T^{2} \) |
| 47 | \( 1 + (-131. - 165. i)T + (-2.31e4 + 1.01e5i)T^{2} \) |
| 53 | \( 1 + (-141. + 619. i)T + (-1.34e5 - 6.45e4i)T^{2} \) |
| 59 | \( 1 - 323.T + 2.05e5T^{2} \) |
| 61 | \( 1 + (-36.8 + 17.7i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 + (223. - 280. i)T + (-6.69e4 - 2.93e5i)T^{2} \) |
| 71 | \( 1 + (209. + 262. i)T + (-7.96e4 + 3.48e5i)T^{2} \) |
| 73 | \( 1 + (15.5 + 68.1i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 + (-350. + 439. i)T + (-1.09e5 - 4.80e5i)T^{2} \) |
| 83 | \( 1 + (1.20e3 + 581. i)T + (3.56e5 + 4.47e5i)T^{2} \) |
| 89 | \( 1 + (-192. + 841. i)T + (-6.35e5 - 3.05e5i)T^{2} \) |
| 97 | \( 1 + (-1.38e3 - 665. i)T + (5.69e5 + 7.13e5i)T^{2} \) |
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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
−10.63381390857357179921288523111, −9.892605401451595943624813031175, −8.824964176845862938138450366463, −7.973807176659559202023198095161, −7.14596870035758954964186890440, −5.88387360446750281991353752541, −4.79520923244230707631983905298, −3.38404622374959264129309267404, −2.26614903859333011567562317450, −0.53575526966578256548907177429,
1.71709043606092447482986936668, 3.03041607935522195952647752164, 4.25807175515000148499170156150, 5.32000427384917021807133186112, 6.71371263766860860028991885581, 7.57351275805894940831499172358, 8.608438750619636367554536365631, 9.587559717796852180660830029476, 10.19774002807579727036807107275, 11.37982117542051677975074681306
