| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 2.00·4-s − 2.23i·5-s + 2.44·6-s + 1.52i·7-s − 2.82·8-s + 2.99·9-s + 3.16i·10-s − 2.20i·11-s − 3.46·12-s − 1.82·13-s − 2.15i·14-s + 3.87i·15-s + 4.00·16-s + 23.2i·17-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.500·4-s − 0.447i·5-s + 0.408·6-s + 0.217i·7-s − 0.353·8-s + 0.333·9-s + 0.316i·10-s − 0.200i·11-s − 0.288·12-s − 0.140·13-s − 0.153i·14-s + 0.258i·15-s + 0.250·16-s + 1.36i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8669152318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8669152318\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (14.5 + 17.8i)T \) |
| good | 7 | \( 1 - 1.52iT - 49T^{2} \) |
| 11 | \( 1 + 2.20iT - 121T^{2} \) |
| 13 | \( 1 + 1.82T + 169T^{2} \) |
| 17 | \( 1 - 23.2iT - 289T^{2} \) |
| 19 | \( 1 + 2.85iT - 361T^{2} \) |
| 29 | \( 1 + 7.23T + 841T^{2} \) |
| 31 | \( 1 - 49.9T + 961T^{2} \) |
| 37 | \( 1 + 9.68iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 62.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 76.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 17.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 33.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 35.2T + 3.48e3T^{2} \) |
| 61 | \( 1 + 8.54iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 78.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 111.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 134.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 72.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 119. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 73.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 178. iT - 9.40e3T^{2} \) |
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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
−10.28660484929789224163199391556, −9.619107492901950472766097756899, −8.484324762623791980282735274053, −8.055335596681974442842807755886, −6.72946003618291675902930007414, −6.07075456842639870887492539000, −5.01671396563295211626890595889, −3.85629247157500504827555506444, −2.27294806911019681147223713478, −0.935851817101448739794560123645,
0.55253120880504243745400735495, 2.10267605197188014379791681504, 3.42429445709044359034393083092, 4.78641925018093022882457596693, 5.82798367261639216618467314863, 6.85761091826139122707899847477, 7.41310939772721801559036471388, 8.431956085509169822402465438992, 9.554550785845657678816809908432, 10.11205369382562532038059919158
