| L(s) = 1 | + (3.14 − 4.26i)2-s + (−1.86 + 0.352i)3-s + (−5.93 − 19.2i)4-s + (−5.08 + 9.95i)5-s + (−4.36 + 9.05i)6-s + (13.8 + 12.3i)7-s + (−60.6 − 21.2i)8-s + (−21.7 + 8.55i)9-s + (26.4 + 53.0i)10-s + (−22.8 + 58.0i)11-s + (17.8 + 33.7i)12-s + (−29.0 − 3.27i)13-s + (96.1 − 20.0i)14-s + (5.96 − 20.3i)15-s + (−148. + 101. i)16-s + (−76.3 − 2.85i)17-s + ⋯ |
| L(s) = 1 | + (1.11 − 1.50i)2-s + (−0.358 + 0.0678i)3-s + (−0.741 − 2.40i)4-s + (−0.455 + 0.890i)5-s + (−0.296 + 0.616i)6-s + (0.745 + 0.666i)7-s + (−2.68 − 0.937i)8-s + (−0.806 + 0.316i)9-s + (0.836 + 1.67i)10-s + (−0.624 + 1.59i)11-s + (0.428 + 0.811i)12-s + (−0.620 − 0.0699i)13-s + (1.83 − 0.382i)14-s + (0.102 − 0.350i)15-s + (−2.32 + 1.58i)16-s + (−1.08 − 0.0407i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.711418 + 0.358975i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.711418 + 0.358975i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (5.08 - 9.95i)T \) |
| 7 | \( 1 + (-13.8 - 12.3i)T \) |
| good | 2 | \( 1 + (-3.14 + 4.26i)T + (-2.35 - 7.64i)T^{2} \) |
| 3 | \( 1 + (1.86 - 0.352i)T + (25.1 - 9.86i)T^{2} \) |
| 11 | \( 1 + (22.8 - 58.0i)T + (-975. - 905. i)T^{2} \) |
| 13 | \( 1 + (29.0 + 3.27i)T + (2.14e3 + 488. i)T^{2} \) |
| 17 | \( 1 + (76.3 + 2.85i)T + (4.89e3 + 367. i)T^{2} \) |
| 19 | \( 1 + (-25.1 + 43.5i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (0.143 + 3.84i)T + (-1.21e4 + 909. i)T^{2} \) |
| 29 | \( 1 + (179. - 40.9i)T + (2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-285. + 165. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (54.9 - 29.0i)T + (2.85e4 - 4.18e4i)T^{2} \) |
| 41 | \( 1 + (-30.0 - 62.4i)T + (-4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-177. - 507. i)T + (-6.21e4 + 4.95e4i)T^{2} \) |
| 47 | \( 1 + (230. + 169. i)T + (3.06e4 + 9.92e4i)T^{2} \) |
| 53 | \( 1 + (577. + 305. i)T + (8.38e4 + 1.23e5i)T^{2} \) |
| 59 | \( 1 + (17.2 + 229. i)T + (-2.03e5 + 3.06e4i)T^{2} \) |
| 61 | \( 1 + (147. - 479. i)T + (-1.87e5 - 1.27e5i)T^{2} \) |
| 67 | \( 1 + (408. + 109. i)T + (2.60e5 + 1.50e5i)T^{2} \) |
| 71 | \( 1 + (132. - 578. i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-258. + 190. i)T + (1.14e5 - 3.71e5i)T^{2} \) |
| 79 | \( 1 + (-387. - 223. i)T + (2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (26.2 + 232. i)T + (-5.57e5 + 1.27e5i)T^{2} \) |
| 89 | \( 1 + (-95.5 - 243. i)T + (-5.16e5 + 4.79e5i)T^{2} \) |
| 97 | \( 1 + (-314. - 314. i)T + 9.12e5iT^{2} \) |
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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
−11.57176500400770020548231444064, −11.28509495670689231053571670054, −10.33819443867279432173444196244, −9.427314145925351737605737709867, −7.81788284074725279273577906302, −6.33090248217513120822464792199, −5.04670383663620326759315634795, −4.49320140574909330538904050262, −2.78148933183338633697049148560, −2.15818080380921405486867403838,
0.21609858938915969118116900597, 3.34910382720368573394285017088, 4.53446188099766843307439648710, 5.33481761716000323028526737199, 6.17971341441521405443487589168, 7.45552402891090135562516485366, 8.250891732252035999781828512718, 8.881447574800847150104722300375, 10.98351456482631636813738403298, 11.81953353776307965602317067727
