| L(s) = 1 | − 4i·2-s + 14.1i·3-s − 16·4-s + (55.0 + 9.77i)5-s + 56.6·6-s + 1.53i·7-s + 64i·8-s + 42.7·9-s + (39.0 − 220. i)10-s + 219.·11-s − 226. i·12-s − 629. i·13-s + 6.13·14-s + (−138. + 778. i)15-s + 256·16-s − 655. i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.907i·3-s − 0.5·4-s + (0.984 + 0.174i)5-s + 0.641·6-s + 0.0118i·7-s + 0.353i·8-s + 0.175·9-s + (0.123 − 0.696i)10-s + 0.546·11-s − 0.453i·12-s − 1.03i·13-s + 0.00836·14-s + (−0.158 + 0.893i)15-s + 0.250·16-s − 0.549i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.174i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.984 + 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.560382160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.560382160\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4iT \) |
| 5 | \( 1 + (-55.0 - 9.77i)T \) |
| 23 | \( 1 + 529iT \) |
| good | 3 | \( 1 - 14.1iT - 243T^{2} \) |
| 7 | \( 1 - 1.53iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 219.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 629. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 655. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 390.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 4.01e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.09e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.92e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.32e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.17e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.79e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 3.28e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.84e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.67e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.10e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 3.98e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.39e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 1.02e5T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.98e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 7.27e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.50e4iT - 8.58e9T^{2} \) |
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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
−10.94157579741695833472155404571, −10.39041805123244468943338362043, −9.543430054397966102576652558270, −8.907721704752439078506767878110, −7.32667649117346245281991722853, −5.85663018142407846138504014655, −4.90242295841091977428143106789, −3.69661032519885222790253396065, −2.50719271439171000495866739409, −1.00475290581825168629064347750,
1.04881353857243152326304210439, 2.09682907588724798974489083948, 4.08620217425593533480026656901, 5.44571865720164461318667894743, 6.50935544385952360859768127663, 7.04793486300743056253115262077, 8.320390719228902513274274733546, 9.269873192157259162223501957863, 10.12750585050154132361937375861, 11.56183369280206897312111363226
