L(s) = 1 | + (2.65 + 4.59i)2-s + (−10.0 + 17.4i)4-s + (−2.78 − 4.81i)5-s − 64.6·8-s + (14.7 − 25.5i)10-s + (−6.95 + 12.0i)11-s − 38.6·13-s + (−90.8 − 157. i)16-s + (−21.7 + 37.6i)17-s + (−54.5 − 94.4i)19-s + 112.·20-s − 73.8·22-s + (−37.4 − 64.8i)23-s + (47.0 − 81.4i)25-s + (−102. − 177. i)26-s + ⋯ |
L(s) = 1 | + (0.938 + 1.62i)2-s + (−1.26 + 2.18i)4-s + (−0.248 − 0.430i)5-s − 2.85·8-s + (0.466 − 0.808i)10-s + (−0.190 + 0.330i)11-s − 0.825·13-s + (−1.41 − 2.45i)16-s + (−0.310 + 0.537i)17-s + (−0.658 − 1.14i)19-s + 1.25·20-s − 0.715·22-s + (−0.339 − 0.587i)23-s + (0.376 − 0.651i)25-s + (−0.774 − 1.34i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.07682458403\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07682458403\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-2.65 - 4.59i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (2.78 + 4.81i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (6.95 - 12.0i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 38.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + (21.7 - 37.6i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (54.5 + 94.4i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (37.4 + 64.8i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 72.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-32.0 + 55.4i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (94.3 + 163. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 24.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 243.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-310. - 537. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (143. - 249. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (262. - 454. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (191. + 332. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (99.0 - 171. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 785.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (165. - 286. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (218. + 379. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 241.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (792. + 1.37e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 79.2T + 9.12e5T^{2} \) |
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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
−11.98415854596321033401882014133, −10.55846842212557921620287929137, −9.177476826975389689986942028176, −8.450445184870413445095199356000, −7.58434200727932744721983163177, −6.76307387627932289189216753165, −5.89025262390120226405936049478, −4.69115469685437902619954024268, −4.33403753550246576633681050488, −2.73216010362568910415006088164,
0.01729676698181807466941693272, 1.65221613018659013127719043014, 2.81756153933936493034272681450, 3.68318422021775237385144339894, 4.76927284604221326219772686261, 5.66302659499996476961097933087, 6.90956343324848268491132223786, 8.351993193804418590144551512443, 9.533694655190003406619735227916, 10.27242316282823050596639763208