| L(s) = 1 | + (0.559 − 0.0736i)2-s + (−0.442 − 0.896i)3-s + (−1.62 + 0.435i)4-s + (1.94 − 1.70i)5-s + (−0.313 − 0.469i)6-s + (2.53 − 0.746i)7-s + (−1.91 + 0.794i)8-s + (−0.608 + 0.793i)9-s + (0.962 − 1.09i)10-s + (0.958 − 0.0627i)11-s + (1.10 + 1.26i)12-s + (−3.77 − 3.77i)13-s + (1.36 − 0.604i)14-s + (−2.39 − 0.990i)15-s + (1.89 − 1.09i)16-s + (3.07 − 2.74i)17-s + ⋯ |
| L(s) = 1 | + (0.395 − 0.0520i)2-s + (−0.255 − 0.517i)3-s + (−0.812 + 0.217i)4-s + (0.870 − 0.763i)5-s + (−0.127 − 0.191i)6-s + (0.959 − 0.282i)7-s + (−0.678 + 0.280i)8-s + (−0.202 + 0.264i)9-s + (0.304 − 0.347i)10-s + (0.288 − 0.0189i)11-s + (0.320 + 0.365i)12-s + (−1.04 − 1.04i)13-s + (0.364 − 0.161i)14-s + (−0.617 − 0.255i)15-s + (0.474 − 0.274i)16-s + (0.745 − 0.666i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.10729 - 0.921882i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10729 - 0.921882i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.442 + 0.896i)T \) |
| 7 | \( 1 + (-2.53 + 0.746i)T \) |
| 17 | \( 1 + (-3.07 + 2.74i)T \) |
| good | 2 | \( 1 + (-0.559 + 0.0736i)T + (1.93 - 0.517i)T^{2} \) |
| 5 | \( 1 + (-1.94 + 1.70i)T + (0.652 - 4.95i)T^{2} \) |
| 11 | \( 1 + (-0.958 + 0.0627i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (3.77 + 3.77i)T + 13iT^{2} \) |
| 19 | \( 1 + (0.417 + 3.16i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (3.58 + 1.76i)T + (14.0 + 18.2i)T^{2} \) |
| 29 | \( 1 + (-0.353 - 1.77i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (-9.23 + 4.55i)T + (18.8 - 24.5i)T^{2} \) |
| 37 | \( 1 + (0.606 - 9.26i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (1.08 - 5.44i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (0.368 + 0.889i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (3.22 - 12.0i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.111 - 0.145i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (11.5 + 1.52i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (-3.35 - 9.88i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (-5.04 - 2.91i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.65 + 2.44i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-9.25 - 3.14i)T + (57.9 + 44.4i)T^{2} \) |
| 79 | \( 1 + (-6.43 + 13.0i)T + (-48.0 - 62.6i)T^{2} \) |
| 83 | \( 1 + (6.07 - 14.6i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-9.93 - 2.66i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-5.25 + 1.04i)T + (89.6 - 37.1i)T^{2} \) |
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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
−11.60857104225244900657931502432, −10.17289402493474488223940943242, −9.459470402731925844363295042295, −8.324045269913900486189297029993, −7.68932171743920569974096855866, −6.14547927587188771535406072842, −5.08415468591289071251532296971, −4.66486536470864084842367950088, −2.76253493598468979025076756122, −1.01014758083278968851419526848,
2.01588348733456223427438672156, 3.73906418882047566434573926080, 4.79490862287624218744694170600, 5.66509447209820807214450730586, 6.53834034088505474039715653807, 8.014080503632672113521244155285, 9.104387982551011987845305492287, 9.948382059863666150426342503709, 10.48350286034594897677555224012, 11.81370819426971325553161795270
