L(s) = 1 | + (1.96 + 4.81i)3-s + (−6.30 + 6.30i)5-s + 24.6·7-s + (−19.3 + 18.8i)9-s + (40.4 + 40.4i)11-s + (47.3 − 47.3i)13-s + (−42.6 − 17.9i)15-s + 41.7i·17-s + (−10.6 − 10.6i)19-s + (48.3 + 118. i)21-s − 53.4i·23-s + 45.5i·25-s + (−128. − 55.9i)27-s + (105. + 105. i)29-s − 3.14i·31-s + ⋯ |
L(s) = 1 | + (0.377 + 0.926i)3-s + (−0.563 + 0.563i)5-s + 1.33·7-s + (−0.715 + 0.698i)9-s + (1.10 + 1.10i)11-s + (1.00 − 1.00i)13-s + (−0.734 − 0.309i)15-s + 0.595i·17-s + (−0.128 − 0.128i)19-s + (0.502 + 1.23i)21-s − 0.484i·23-s + 0.364i·25-s + (−0.917 − 0.398i)27-s + (0.674 + 0.674i)29-s − 0.0182i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.341 - 0.939i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.341 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.337265302\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.337265302\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.96 - 4.81i)T \) |
good | 5 | \( 1 + (6.30 - 6.30i)T - 125iT^{2} \) |
| 7 | \( 1 - 24.6T + 343T^{2} \) |
| 11 | \( 1 + (-40.4 - 40.4i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 + (-47.3 + 47.3i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 - 41.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (10.6 + 10.6i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 + 53.4iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-105. - 105. i)T + 2.43e4iT^{2} \) |
| 31 | \( 1 + 3.14iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (42.1 + 42.1i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 - 152.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (221. - 221. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + 381.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (294. - 294. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (445. + 445. i)T + 2.05e5iT^{2} \) |
| 61 | \( 1 + (21.8 - 21.8i)T - 2.26e5iT^{2} \) |
| 67 | \( 1 + (-572. - 572. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 + 612. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 331. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 427. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-245. + 245. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + 188.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.47e3T + 9.12e5T^{2} \) |
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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
−11.01963276598452362907609216022, −10.49054760695504557832103508738, −9.342107561211348665500170187906, −8.363763623011891048844101876124, −7.75384523762054189797638639021, −6.44852602976066082911309593699, −5.04937140223686474700390082266, −4.20821621810785237032990185409, −3.23020015598247748409596589133, −1.61458538009387961812033604819,
0.847514773192127255546944145091, 1.76483003705546352967757722483, 3.51572105944084610269373455548, 4.57642629187475559476256341729, 5.97734851113181608436141829078, 6.89294376479595149842768204017, 8.166746691272321850012543680695, 8.449495398874918084853779706939, 9.331127423847103019455463913799, 11.15534595380159661011884369798