L(s) = 1 | + 1.41i·5-s + 4i·7-s − 5.65·11-s + 4·13-s + 4.24i·17-s + 5.65·23-s + 2.99·25-s − 1.41i·29-s + 4i·31-s − 5.65·35-s − 6·37-s − 9.89i·41-s − 8i·43-s + 5.65·47-s − 9·49-s + ⋯ |
L(s) = 1 | + 0.632i·5-s + 1.51i·7-s − 1.70·11-s + 1.10·13-s + 1.02i·17-s + 1.17·23-s + 0.599·25-s − 0.262i·29-s + 0.718i·31-s − 0.956·35-s − 0.986·37-s − 1.54i·41-s − 1.21i·43-s + 0.825·47-s − 1.28·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.886511 + 0.747364i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.886511 + 0.747364i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.41iT - 5T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 4.24iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 9.89iT - 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 - 4.24iT - 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 - 5.65T + 83T^{2} \) |
| 89 | \( 1 + 4.24iT - 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−12.10557899429009744748115198641, −10.82456005537774472019773432867, −10.49308058682219561740548930111, −8.926482804536720712728402979001, −8.405246948675098042165002227671, −7.12286463458675843000321819137, −5.91809348584164147656764359401, −5.19094359376972883526144959599, −3.34792848021892115755775602456, −2.27276346214013939093115467313,
0.913998941652980752511952537535, 3.06009488420297902369055896034, 4.46075670744772068972787034508, 5.35211201670349974964006866122, 6.85672940463058260962364787398, 7.72772175526932976034699506303, 8.642666308164373347887580215638, 9.850913137305034046931612556987, 10.71247918966201370982923165129, 11.36080470598224067289927227618