| L(s) = 1 | − 3-s − 3.46i·5-s + 9-s − 3.46i·11-s + (−1 − 3.46i)13-s + 3.46i·15-s + 6·17-s + 6.92i·19-s − 6.99·25-s − 27-s − 6·29-s + 6.92i·31-s + 3.46i·33-s + (1 + 3.46i)39-s + 3.46i·41-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.54i·5-s + 0.333·9-s − 1.04i·11-s + (−0.277 − 0.960i)13-s + 0.894i·15-s + 1.45·17-s + 1.58i·19-s − 1.39·25-s − 0.192·27-s − 1.11·29-s + 1.24i·31-s + 0.603i·33-s + (0.160 + 0.554i)39-s + 0.541i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.734960 - 0.552806i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.734960 - 0.552806i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + (1 + 3.46i)T \) |
| good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 6.92iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 6.92iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 3.46iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 3.46iT - 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 13.8iT - 67T^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 3.46iT - 83T^{2} \) |
| 89 | \( 1 - 17.3iT - 89T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
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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
−12.47777993442475047537149674355, −12.09610087274707295390961533868, −10.71055167168886695063167322870, −9.716465742809212230492610989670, −8.521078990654490500488221615857, −7.70741301796249880192626477598, −5.77786494731870035702275547812, −5.30027144236001040420495900093, −3.70099785460390951534581930114, −1.05917143047944965551776844665,
2.45404452071008238939658620604, 4.11773132345347272338645197902, 5.66781894923398035828050631991, 6.97033036257582538033588600299, 7.40068433771956951407573032606, 9.386613936137468690359682807859, 10.21490113531554331368334859418, 11.20432290249664774404681518217, 11.88890078798112065973412722504, 13.10698815434128974986256965818
