| L(s) = 1 | + (0.0152 − 0.00133i)2-s + (−1.96 + 0.347i)4-s + (1.79 − 1.33i)5-s + (0.211 − 0.148i)7-s + (−0.0589 + 0.0158i)8-s + (0.0255 − 0.0226i)10-s + (−1.49 − 4.11i)11-s + (0.187 − 2.14i)13-s + (0.00301 − 0.00253i)14-s + (3.75 − 1.36i)16-s + (−1.70 − 0.456i)17-s + (4.91 + 2.83i)19-s + (−3.06 + 3.25i)20-s + (−0.0282 − 0.0605i)22-s + (3.32 − 4.74i)23-s + ⋯ |
| L(s) = 1 | + (0.0107 − 0.000940i)2-s + (−0.984 + 0.173i)4-s + (0.802 − 0.596i)5-s + (0.0799 − 0.0559i)7-s + (−0.0208 + 0.00558i)8-s + (0.00806 − 0.00717i)10-s + (−0.451 − 1.23i)11-s + (0.0519 − 0.594i)13-s + (0.000806 − 0.000676i)14-s + (0.939 − 0.341i)16-s + (−0.412 − 0.110i)17-s + (1.12 + 0.651i)19-s + (−0.686 + 0.727i)20-s + (−0.00601 − 0.0129i)22-s + (0.693 − 0.989i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.970005 - 0.650613i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.970005 - 0.650613i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.79 + 1.33i)T \) |
| good | 2 | \( 1 + (-0.0152 + 0.00133i)T + (1.96 - 0.347i)T^{2} \) |
| 7 | \( 1 + (-0.211 + 0.148i)T + (2.39 - 6.57i)T^{2} \) |
| 11 | \( 1 + (1.49 + 4.11i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.187 + 2.14i)T + (-12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (1.70 + 0.456i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-4.91 - 2.83i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.32 + 4.74i)T + (-7.86 - 21.6i)T^{2} \) |
| 29 | \( 1 + (3.59 + 3.01i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.912 + 5.17i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (0.837 - 3.12i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (0.241 + 0.288i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.84 + 8.25i)T + (-27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (-2.29 - 3.28i)T + (-16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (-8.15 - 8.15i)T + 53iT^{2} \) |
| 59 | \( 1 + (10.6 + 3.86i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (2.10 - 11.9i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.82 - 0.160i)T + (65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (4.44 - 2.56i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.71 - 13.8i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.98 + 2.37i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.432 - 4.94i)T + (-81.7 + 14.4i)T^{2} \) |
| 89 | \( 1 + (-1.44 + 2.50i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.27 + 1.99i)T + (62.3 + 74.3i)T^{2} \) |
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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
−10.93853824974532208365491633303, −10.05900173074457708366173101165, −9.165410469360617553172994709547, −8.509887703605462983906559084343, −7.61582297273030522601915853096, −5.91965773192701788223175089374, −5.38625881495130920522718092042, −4.22199215023551456195996302684, −2.87204742447380718936998715550, −0.836872879079531810798887576801,
1.77346081509742684541433016099, 3.32792127684363019341922616540, 4.77684288052826768601096985221, 5.44977470031089115676618959757, 6.78486064291313070112335821027, 7.60755925854980515299392007369, 9.090746656284520158728110070285, 9.475895689691754641730473773565, 10.36599060959774239284467987159, 11.27807766308572291881528792626
