| L(s) = 1 | − 20.6i·2-s − 63.1·3-s − 299.·4-s − 321.·5-s + 1.30e3i·6-s + 710.·7-s + 3.54e3i·8-s + 1.80e3·9-s + 6.65e3i·10-s − 1.19e3i·11-s + 1.89e4·12-s − 1.57e3i·13-s − 1.46e4i·14-s + 2.03e4·15-s + 3.49e4·16-s + 1.83e4i·17-s + ⋯ |
| L(s) = 1 | − 1.82i·2-s − 1.35·3-s − 2.33·4-s − 1.15·5-s + 2.46i·6-s + 0.782·7-s + 2.44i·8-s + 0.824·9-s + 2.10i·10-s − 0.270i·11-s + 3.16·12-s − 0.198i·13-s − 1.43i·14-s + 1.55·15-s + 2.13·16-s + 0.907i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.423i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.905 + 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.4889155480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4889155480\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 109 | \( 1 + (1.22e7 - 5.73e6i)T \) |
| good | 2 | \( 1 + 20.6iT - 128T^{2} \) |
| 3 | \( 1 + 63.1T + 2.18e3T^{2} \) |
| 5 | \( 1 + 321.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 710.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.19e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 1.57e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 1.83e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 6.83e3iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 7.99e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 3.52e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.99e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.48e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 6.32e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 7.73e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.16e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 1.11e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 1.10e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 1.25e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.41e5iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 2.92e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.25e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.77e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 5.49e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.82e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.43e7T + 8.07e13T^{2} \) |
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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
−11.64777568912225702956421789334, −11.10928345377536875870062938404, −10.41585666250696830808580460794, −8.912441314122957521200456709059, −7.66500753990616742407452838244, −5.60721982990721809650899762938, −4.51205770583046592114233718741, −3.49341943974011357429789226522, −1.58896154118505213782010905044, −0.39805236975375989723456775797,
0.54637001343680928234982696653, 4.26305689978117284478161453083, 4.99048242769280238124996926133, 6.07514270702026305794522314238, 7.18720258156978394853439885257, 7.902581669895927425063662944560, 9.116273011430054120855700710375, 10.79645156750361366653339774859, 11.76600144743616231789056596154, 12.74742778754142590551560054189
