L(s) = 1 | + i·2-s + 0.287i·3-s − 4-s + (−0.437 − 2.19i)5-s − 0.287·6-s − 1.04i·7-s − i·8-s + 2.91·9-s + (2.19 − 0.437i)10-s + 3.63·11-s − 0.287i·12-s − 1.41i·13-s + 1.04·14-s + (0.630 − 0.125i)15-s + 16-s + 3.91i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.165i·3-s − 0.5·4-s + (−0.195 − 0.980i)5-s − 0.117·6-s − 0.394i·7-s − 0.353i·8-s + 0.972·9-s + (0.693 − 0.138i)10-s + 1.09·11-s − 0.0829i·12-s − 0.391i·13-s + 0.278·14-s + (0.162 − 0.0324i)15-s + 0.250·16-s + 0.950i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.31635 + 0.129939i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31635 + 0.129939i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (0.437 + 2.19i)T \) |
| 31 | \( 1 - T \) |
good | 3 | \( 1 - 0.287iT - 3T^{2} \) |
| 7 | \( 1 + 1.04iT - 7T^{2} \) |
| 11 | \( 1 - 3.63T + 11T^{2} \) |
| 13 | \( 1 + 1.41iT - 13T^{2} \) |
| 17 | \( 1 - 3.91iT - 17T^{2} \) |
| 19 | \( 1 - 4.96T + 19T^{2} \) |
| 23 | \( 1 + 8.00iT - 23T^{2} \) |
| 29 | \( 1 + 6.59T + 29T^{2} \) |
| 37 | \( 1 - 3.71iT - 37T^{2} \) |
| 41 | \( 1 - 2.74T + 41T^{2} \) |
| 43 | \( 1 + 3.73iT - 43T^{2} \) |
| 47 | \( 1 - 2.35iT - 47T^{2} \) |
| 53 | \( 1 + 0.335iT - 53T^{2} \) |
| 59 | \( 1 + 4.33T + 59T^{2} \) |
| 61 | \( 1 + 8.33T + 61T^{2} \) |
| 67 | \( 1 - 8.27iT - 67T^{2} \) |
| 71 | \( 1 + 5.59T + 71T^{2} \) |
| 73 | \( 1 - 8.43iT - 73T^{2} \) |
| 79 | \( 1 - 1.42T + 79T^{2} \) |
| 83 | \( 1 + 12.8iT - 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 - 16.4iT - 97T^{2} \) |
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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
−11.95804427361975966894425130204, −10.60091069159777763543910825235, −9.637682883163733632887451167865, −8.843643615556759329297684210013, −7.86428643961515445002350242723, −6.93063592659653336043272082075, −5.78347897089306501919210248860, −4.56023725220874372782715110565, −3.84655997027808174360021860771, −1.21209082254864305147923445681,
1.65444680949152267990034502378, 3.16951795692188074473803437244, 4.17779458516582549589139341537, 5.67268527056690203227759020433, 6.98884610064339837903991823269, 7.63691473210040009973289761050, 9.365351989279441715349952440821, 9.609576274029393517041302991530, 10.94880652361043597141386902734, 11.64472432760279515291649857378