L(s) = 1 | + (−0.866 + 0.5i)7-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)13-s + i·17-s + 19-s + (−0.866 − 0.5i)23-s + (0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + i·37-s + (−0.5 + 0.866i)41-s + (0.866 − 0.5i)43-s + (−0.866 + 0.5i)47-s + (0.5 − 0.866i)49-s − i·53-s + (−0.5 + 0.866i)59-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)7-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)13-s + i·17-s + 19-s + (−0.866 − 0.5i)23-s + (0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + i·37-s + (−0.5 + 0.866i)41-s + (0.866 − 0.5i)43-s + (−0.866 + 0.5i)47-s + (0.5 − 0.866i)49-s − i·53-s + (−0.5 + 0.866i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.619 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.619 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9495318916 + 0.4602231491i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9495318916 + 0.4602231491i\) |
\(L(1)\) |
\(\approx\) |
\(0.9932061050 + 0.2006857908i\) |
\(L(1)\) |
\(\approx\) |
\(0.9932061050 + 0.2006857908i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.866 + 0.5i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.866 + 0.5i)T \) |
| 17 | \( 1 + iT \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (-0.866 - 0.5i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (0.866 - 0.5i)T \) |
| 47 | \( 1 + (-0.866 + 0.5i)T \) |
| 53 | \( 1 - iT \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.866 + 0.5i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.866 - 0.5i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (0.866 - 0.5i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−27.06822445086163963938830700733, −26.31311123283726537661806409912, −25.2626295545272811545971103270, −24.423537746949986519351368212937, −23.16553773004253715806186175541, −22.56239738322212448812516435936, −21.45151319133627261246892573044, −20.31105519360159571983256243395, −19.53184149208388347242844865690, −18.48387437286542438942711141644, −17.453156410996926075463308551530, −16.17482144336132319399123562679, −15.7829784462555658009346478554, −14.007658176734994785593692744351, −13.570771950784608454571089218743, −12.222292661173538327023009411992, −11.16832162846060285637782555208, −10.02937177502356290641910606957, −9.03803818027556563307862346944, −7.76623829328957830807320725434, −6.564411785746478074097098740411, −5.56210090651957534747492329506, −3.9095395532430418668307301454, −2.975344835202303108590410400202, −0.938233381547642168904106781351,
1.66565726402472241815361418806, 3.20159152874674930796969735462, 4.40214035112962352645537886000, 5.95944605187196083986686695306, 6.76766448682723434216717310651, 8.23162708800077380170417330891, 9.342582437927027999059610857336, 10.22362519358324364568175750206, 11.64758204806397681604212559287, 12.509286126567353094034239841438, 13.55410121485110522875064761202, 14.73242064784282077055427454335, 15.75266391419677700459798655526, 16.59894760737686537576120890115, 17.82406890398442171492886473654, 18.74628050463900670532404041834, 19.72864575300969468685817053808, 20.63799047934211511777389455934, 21.89091841068258405995197363219, 22.56382180121813583667587860308, 23.59902782394974141978439881327, 24.662894084338377581195659487087, 25.7422062627983920615409643532, 26.19821677793104865592344914207, 27.66857156291805264024609153752