L(s) = 1 | + 3.41i·5-s + 4.82i·7-s + 2.82·11-s + 2.82·13-s − 5.41i·17-s + 5.65i·19-s − 1.17·23-s − 6.65·25-s + 0.585i·29-s − 3.17i·31-s − 16.4·35-s + 3.65·37-s − 2.58i·41-s + 9.65i·43-s − 12.4·47-s + ⋯ |
L(s) = 1 | + 1.52i·5-s + 1.82i·7-s + 0.852·11-s + 0.784·13-s − 1.31i·17-s + 1.29i·19-s − 0.244·23-s − 1.33·25-s + 0.108i·29-s − 0.569i·31-s − 2.78·35-s + 0.601·37-s − 0.403i·41-s + 1.47i·43-s − 1.82·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.616177962\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.616177962\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 3.41iT - 5T^{2} \) |
| 7 | \( 1 - 4.82iT - 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 5.41iT - 17T^{2} \) |
| 19 | \( 1 - 5.65iT - 19T^{2} \) |
| 23 | \( 1 + 1.17T + 23T^{2} \) |
| 29 | \( 1 - 0.585iT - 29T^{2} \) |
| 31 | \( 1 + 3.17iT - 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 + 2.58iT - 41T^{2} \) |
| 43 | \( 1 - 9.65iT - 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 5.07iT - 53T^{2} \) |
| 59 | \( 1 + 2.34T + 59T^{2} \) |
| 61 | \( 1 - 7.65T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 4.48T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 6.48iT - 79T^{2} \) |
| 83 | \( 1 - 5.17T + 83T^{2} \) |
| 89 | \( 1 + 12.2iT - 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−9.893816773756206006152652069949, −9.416024848594289385158154012154, −8.443983538437533135179292021629, −7.63976711841726707390573560959, −6.38429287030864020589896875492, −6.23282334091230663456225298784, −5.12113368635456465437296374761, −3.64822522226874651074227129262, −2.88210380728742240201039710144, −1.91805364102669268549813230830,
0.75882814927874498073317092590, 1.54190869856130659540332771300, 3.65527485532749418032713803890, 4.22017826406989044631952144554, 5.00029805967431548031282124158, 6.23560834969160719434475012130, 6.99974915433235977548441406420, 8.039592069650999901534996369624, 8.652408617807454950946310375541, 9.437706038300084413516386713056