L(s) = 1 | + (0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s − 13-s + 15-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s − 27-s − 29-s + (−0.5 + 0.866i)31-s + (−0.5 − 0.866i)33-s + (0.5 + 0.866i)37-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s − 13-s + 15-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s − 27-s − 29-s + (−0.5 + 0.866i)31-s + (−0.5 − 0.866i)33-s + (0.5 + 0.866i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.015749404 - 0.2379507569i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.015749404 - 0.2379507569i\) |
\(L(1)\) |
\(\approx\) |
\(1.144416812 - 0.1890773524i\) |
\(L(1)\) |
\(\approx\) |
\(1.144416812 - 0.1890773524i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 + (0.5 - 0.866i)T \) |
| 59 | \( 1 + (0.5 - 0.866i)T \) |
| 61 | \( 1 + (0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−33.063039205310755923098035665845, −32.03544889421736177114128467462, −31.22745060950545181521565468473, −29.6928220809198263794187545187, −28.377901088678679940072714335614, −27.57475672962583227682316732842, −26.32609648903301844075867008414, −25.22428274137713926686550752869, −24.31433937438968621315748409359, −22.515172534202875348241627428914, −21.60931796040119349508270047159, −20.35104712810075107830630665127, −19.77054595273005960804550521510, −17.73802521766610363020169011157, −16.69457854557393395856180513184, −15.516200172712180204782466577034, −14.30072552858230477095100671407, −13.05736614935335163935275630588, −11.51874359844929761671569982875, −9.697367100622115517238410527, −9.22939246937515407603293390851, −7.5001379108129929193308950465, −5.34514023322726001166668039247, −4.275643771230588155053918649108, −2.286932854124725017211133938959,
1.979187922552712419455905670158, 3.42657634915339302822986887042, 5.93914354997256888939485106431, 7.02960834117298962590599781846, 8.421353673615397191135210682863, 9.9242602144380768221004404504, 11.47394543335907408927272641089, 12.86403045089641002680052356428, 14.13726659162291285573375640509, 14.82168300855820107816928743341, 16.83173382948516944677621157313, 18.067126830961526936755658746915, 18.99002657639154825336832760314, 20.019811095619857066711058797069, 21.59229649401529206561549190091, 22.626027887732602347617102642830, 24.119636756062502195205479758392, 24.93777312522420219237719763657, 26.14801703818197893491223395759, 26.96942614732156147411361702982, 28.88164856972576021512460322664, 29.711538087543605866881860291912, 30.54313326014748863362206707886, 31.6858212545645373072385833513, 32.878652151155462845550774415780