L(s) = 1 | + (−0.5 − 0.866i)5-s + (−0.866 + 1.5i)7-s + (0.866 + 1.5i)23-s + (−0.499 + 0.866i)25-s + (−0.5 + 0.866i)29-s + 1.73·35-s + (0.5 + 0.866i)41-s + (−0.866 + 1.5i)47-s + (−1 − 1.73i)49-s + (−0.5 + 0.866i)61-s + (0.866 + 1.5i)67-s + (0.866 − 1.5i)83-s + 89-s + (1 − 1.73i)101-s − 1.73·107-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)5-s + (−0.866 + 1.5i)7-s + (0.866 + 1.5i)23-s + (−0.499 + 0.866i)25-s + (−0.5 + 0.866i)29-s + 1.73·35-s + (0.5 + 0.866i)41-s + (−0.866 + 1.5i)47-s + (−1 − 1.73i)49-s + (−0.5 + 0.866i)61-s + (0.866 + 1.5i)67-s + (0.866 − 1.5i)83-s + 89-s + (1 − 1.73i)101-s − 1.73·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7671400697\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7671400697\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
good | 7 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.224665982574110897268308440571, −8.880721768977359384084929873906, −7.985956267914304075923238843884, −7.17822070982291029772499092537, −6.12640977067959197527110102804, −5.48617037245777466407456636448, −4.76348182084597636064235446332, −3.57165095179946575183583520492, −2.81541551353008979559568852167, −1.48127986216760581996466882832,
0.55141582021464210397343619677, 2.38439582429377656778591876945, 3.45474718625286887030640649981, 3.95832788026268393827972234958, 4.94316220882254008543057175965, 6.36389806479051389419857020897, 6.70061787975834571015686303652, 7.46621794624362066218790221929, 8.103574727267180104440215752817, 9.207830977810224538605690992059