| L(s) = 1 | + (24.5 − 11.2i)3-s − 414. i·7-s + (475. − 552. i)9-s + 61.5i·11-s + 1.37e3i·13-s + 6.80e3·17-s + 7.82e3·19-s + (−4.66e3 − 1.01e4i)21-s + 7.45e3·23-s + (5.47e3 − 1.89e4i)27-s − 424. i·29-s − 4.96e4·31-s + (692. + 1.51e3i)33-s + 2.27e3i·37-s + (1.55e4 + 3.38e4i)39-s + ⋯ |
| L(s) = 1 | + (0.909 − 0.416i)3-s − 1.20i·7-s + (0.652 − 0.757i)9-s + 0.0462i·11-s + 0.627i·13-s + 1.38·17-s + 1.14·19-s + (−0.503 − 1.09i)21-s + 0.613·23-s + (0.277 − 0.960i)27-s − 0.0173i·29-s − 1.66·31-s + (0.0192 + 0.0420i)33-s + 0.0448i·37-s + (0.261 + 0.570i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0339 + 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0339 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.165728784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.165728784\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-24.5 + 11.2i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 414. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 61.5iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 1.37e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 6.80e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 7.82e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 7.45e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + 424. iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 4.96e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 2.27e3iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 5.62e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.31e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 8.47e4T + 1.07e10T^{2} \) |
| 53 | \( 1 - 1.66e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 4.00e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 8.36e4T + 5.15e10T^{2} \) |
| 67 | \( 1 - 4.25e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 3.46e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 1.71e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 2.66e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 1.01e6T + 3.26e11T^{2} \) |
| 89 | \( 1 + 9.55e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.19e6iT - 8.32e11T^{2} \) |
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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
−10.25803964633839724535815710854, −9.538461912371211364471249732092, −8.514478081699756492133592354006, −7.33195060985318465498877440248, −7.10118200351371231956689892470, −5.46074404776678603689760134890, −3.99766999933901628524235908435, −3.25046175338289762887732877415, −1.71920539681120372683271813013, −0.71206894580728741394329521073,
1.35271152931887286452293609033, 2.73648311806624216619498503057, 3.44408428928475688466763234045, 5.00332459177721621045281445543, 5.77977006208875557053015908698, 7.40180774534393925049089094963, 8.151816067299315140649687269861, 9.163026201933268744549929692668, 9.736044533723983767478298993164, 10.82921698621530460487407418553
