L(s) = 1 | + (1.18 − 1.18i)2-s + (−5.11 − 0.932i)3-s + 5.17i·4-s + (−7.17 + 4.96i)6-s + (13.3 + 13.3i)7-s + (15.6 + 15.6i)8-s + (25.2 + 9.53i)9-s + 28.7i·11-s + (4.83 − 26.4i)12-s + (−14.1 + 14.1i)13-s + 31.7·14-s − 4.25·16-s + (18.5 − 18.5i)17-s + (41.3 − 18.6i)18-s + 49.0i·19-s + ⋯ |
L(s) = 1 | + (0.419 − 0.419i)2-s + (−0.983 − 0.179i)3-s + 0.647i·4-s + (−0.488 + 0.337i)6-s + (0.721 + 0.721i)7-s + (0.691 + 0.691i)8-s + (0.935 + 0.353i)9-s + 0.787i·11-s + (0.116 − 0.636i)12-s + (−0.302 + 0.302i)13-s + 0.605·14-s − 0.0664·16-s + (0.264 − 0.264i)17-s + (0.541 − 0.244i)18-s + 0.592i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.26327 + 0.561660i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26327 + 0.561660i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (5.11 + 0.932i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-1.18 + 1.18i)T - 8iT^{2} \) |
| 7 | \( 1 + (-13.3 - 13.3i)T + 343iT^{2} \) |
| 11 | \( 1 - 28.7iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (14.1 - 14.1i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + (-18.5 + 18.5i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 - 49.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (37.7 + 37.7i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + 125.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 247.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-127. - 127. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 390. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-39.3 + 39.3i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (-124. + 124. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + (160. + 160. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 - 729.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 2T + 2.26e5T^{2} \) |
| 67 | \( 1 + (-329. - 329. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 - 171. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-279. + 279. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 - 48.0iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-144. - 144. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (908. + 908. i)T + 9.12e5iT^{2} \) |
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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
−13.97119411821330236901440869674, −12.68670275420264609907367151985, −12.02373978637747555566785448863, −11.33386939690620510955120147810, −9.993063827041157420299608153749, −8.274043355187876314353592037486, −7.09342904551073333876800953753, −5.41637500864357446328764901593, −4.30545596656865106666571456559, −2.08934525185872362915185601182,
0.958252888655472075453266712888, 4.26463616948453317763866008823, 5.36383442794152895174637094167, 6.44486560968917519031586572355, 7.73330423690636504857384738042, 9.701343277851257882377132293866, 10.72255311105718620653245219052, 11.46858564047937941168916856938, 12.99018743551617826000168113890, 13.97922697436563668856020275813