L(s) = 1 | + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (−0.587 − 0.809i)7-s + (−0.587 − 0.809i)8-s + (0.951 − 0.309i)9-s + (0.309 − 0.951i)11-s + (0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s − 0.999·18-s + (−0.587 + 0.809i)22-s − 1.61·23-s + (−0.587 + 0.809i)25-s − 0.999i·28-s + (0.278 − 1.76i)29-s − i·32-s + ⋯ |
L(s) = 1 | + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (−0.587 − 0.809i)7-s + (−0.587 − 0.809i)8-s + (0.951 − 0.309i)9-s + (0.309 − 0.951i)11-s + (0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s − 0.999·18-s + (−0.587 + 0.809i)22-s − 1.61·23-s + (−0.587 + 0.809i)25-s − 0.999i·28-s + (0.278 − 1.76i)29-s − i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6511467010\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6511467010\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.951 + 0.309i)T \) |
| 7 | \( 1 + (0.587 + 0.809i)T \) |
| 11 | \( 1 + (-0.309 + 0.951i)T \) |
good | 3 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 5 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 13 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 17 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 23 | \( 1 + 1.61T + T^{2} \) |
| 29 | \( 1 + (-0.278 + 1.76i)T + (-0.951 - 0.309i)T^{2} \) |
| 31 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.0489 + 0.309i)T + (-0.951 - 0.309i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + (-1.26 + 1.26i)T - iT^{2} \) |
| 47 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.809 - 0.412i)T + (0.587 + 0.809i)T^{2} \) |
| 59 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 61 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 67 | \( 1 + (-0.642 + 0.642i)T - iT^{2} \) |
| 71 | \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.809 - 0.587i)T^{2} \) |
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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
−9.745149631023169869555397255260, −9.135025513444030756172616407734, −8.029878360886762815891007615180, −7.47451901027952399421972385690, −6.54979256389811062788129360924, −5.90253607668828589766633811633, −4.05384698038450703274810934243, −3.63663880643696082644406310147, −2.19149708204248676182715529827, −0.798839813504973478374971980620,
1.64155345463761609303613665032, 2.57639248917550033800943596734, 4.08161489722350164908864359297, 5.22914503492577148879516587024, 6.21617415968002057559155486577, 6.88378214072705860151327755749, 7.70113846122415444775398663230, 8.505955720342526902482732752515, 9.407296095266355721160227335253, 9.936441051772792128800911380315