| L(s) = 1 | + (−0.459 − 0.134i)2-s + (0.654 − 0.755i)3-s + (−1.48 − 0.957i)4-s + (0.480 − 3.34i)5-s + (−0.402 + 0.258i)6-s + (1.88 + 4.12i)7-s + (1.18 + 1.36i)8-s + (−0.142 − 0.989i)9-s + (−0.671 + 1.47i)10-s + (−0.942 + 0.276i)11-s + (−1.69 + 0.498i)12-s + (−0.188 + 0.412i)13-s + (−0.309 − 2.15i)14-s + (−2.21 − 2.55i)15-s + (1.11 + 2.43i)16-s + (−2.76 + 1.77i)17-s + ⋯ |
| L(s) = 1 | + (−0.324 − 0.0954i)2-s + (0.378 − 0.436i)3-s + (−0.744 − 0.478i)4-s + (0.214 − 1.49i)5-s + (−0.164 + 0.105i)6-s + (0.712 + 1.56i)7-s + (0.418 + 0.482i)8-s + (−0.0474 − 0.329i)9-s + (−0.212 + 0.465i)10-s + (−0.284 + 0.0834i)11-s + (−0.490 + 0.144i)12-s + (−0.0522 + 0.114i)13-s + (−0.0826 − 0.574i)14-s + (−0.571 − 0.658i)15-s + (0.277 + 0.608i)16-s + (−0.670 + 0.430i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.716207 - 0.383022i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.716207 - 0.383022i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (-4.59 - 1.37i)T \) |
| good | 2 | \( 1 + (0.459 + 0.134i)T + (1.68 + 1.08i)T^{2} \) |
| 5 | \( 1 + (-0.480 + 3.34i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.88 - 4.12i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (0.942 - 0.276i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (0.188 - 0.412i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (2.76 - 1.77i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-1.41 - 0.906i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (1.15 - 0.740i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (4.23 + 4.88i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.506 - 3.52i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.821 - 5.71i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-7.20 + 8.31i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + (5.11 + 11.1i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (4.41 - 9.67i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (3.65 + 4.21i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-7.43 - 2.18i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (2.86 + 0.842i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (3.27 + 2.10i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (3.61 - 7.92i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.06 - 7.40i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-4.73 + 5.46i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.801 + 5.57i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−14.59317806454473233764768091684, −13.27760966795622158288230354159, −12.63050836031843215670832767168, −11.39472602886356109926092421684, −9.511108622896911654984903569732, −8.832205250647139032744721073497, −8.131106538266026302459548322854, −5.68749855274324005236971184280, −4.79844477401519782716514577998, −1.76343600730618198922411243118,
3.27533312150018725849889219813, 4.62869899383640465896629418246, 7.03076364467454043961269197515, 7.77747424201149196099697067312, 9.299755387608431515627816758365, 10.50584691920679389708269758240, 11.05061713956643066350929713815, 13.14531750223005673582843923809, 14.05342964033122422968136298230, 14.58238240085763523732135642459
