L(s) = 1 | − 2·4-s − 2.23i·5-s + 5.38i·11-s + 4·16-s + 4.47i·20-s + 2.23i·23-s − 5.00·25-s − 5.38i·29-s + 12.0·37-s + 5.38i·41-s + 12.0·43-s − 10.7i·44-s + 7·49-s − 11.1i·53-s + 12.0·55-s + ⋯ |
L(s) = 1 | − 4-s − 0.999i·5-s + 1.62i·11-s + 16-s + 0.999i·20-s + 0.466i·23-s − 1.00·25-s − 0.999i·29-s + 1.97·37-s + 0.841i·41-s + 1.83·43-s − 1.62i·44-s + 49-s − 1.53i·53-s + 1.62·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.209199573\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.209199573\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 29 | \( 1 + 5.38iT \) |
good | 2 | \( 1 + 2T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 5.38iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 2.23iT - 23T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 12.0T + 37T^{2} \) |
| 41 | \( 1 - 5.38iT - 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 11.1iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 15.6iT - 83T^{2} \) |
| 89 | \( 1 + 10.7iT - 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−9.633707447013071776654572544945, −8.993300913265241440035247111510, −8.018932023320081428671109150551, −7.50530552887841372903272852926, −6.16531467087960261605465615692, −5.22943269081314089635970392005, −4.50528740654116124079001807500, −3.93652489556129409685887195005, −2.25347376882813811470605930560, −0.895306534938409028171532575126,
0.76028464924005956211439837042, 2.66394786707359359663027293418, 3.52605629396906719909437971818, 4.37871389256142542574020829343, 5.65496163549387476363382516234, 6.12100844234200891893254203097, 7.30089514700373161843594391373, 8.088307747936272021660608978913, 8.894354164122110518402517617050, 9.525580982329992271217552998696