L(s) = 1 | + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.707 + 0.707i)5-s + (0.5 + 1.86i)7-s + (0.707 − 0.707i)8-s + (−0.500 + 0.866i)10-s + (−0.448 − 0.258i)11-s + (0.965 + 1.67i)14-s + (0.500 − 0.866i)16-s + (−0.258 + 0.965i)20-s + (−0.5 − 0.133i)22-s − 1.00i·25-s + (1.36 + 1.36i)28-s + (−0.707 + 1.22i)29-s + (0.866 + 1.5i)31-s + (0.258 − 0.965i)32-s + ⋯ |
L(s) = 1 | + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.707 + 0.707i)5-s + (0.5 + 1.86i)7-s + (0.707 − 0.707i)8-s + (−0.500 + 0.866i)10-s + (−0.448 − 0.258i)11-s + (0.965 + 1.67i)14-s + (0.500 − 0.866i)16-s + (−0.258 + 0.965i)20-s + (−0.5 − 0.133i)22-s − 1.00i·25-s + (1.36 + 1.36i)28-s + (−0.707 + 1.22i)29-s + (0.866 + 1.5i)31-s + (0.258 − 0.965i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.099381615\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.099381615\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 59 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \) |
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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
−8.722795913856955874064510477884, −8.254173030771882922484232560837, −7.24132880189635478637979103549, −6.59849134894058783708611038094, −5.64378823554655597838170503466, −5.24282981437002069310966770922, −4.30287932573940244888826633189, −3.18533247167359960932111842068, −2.73374414430836611974299332766, −1.75142085898235685688368109043,
0.952448719447704570229869862896, 2.27215279828599388762423696688, 3.64046150644504207436401518457, 4.14438396768137923666939217203, 4.67781382477935453270461620346, 5.50302707381867484531293318819, 6.53802048633187411484510705582, 7.35312017348573635329058720656, 7.81156949554493737773756060467, 8.250615048819885910081971967484