| L(s) = 1 | − 7.34i·5-s − 10.3·7-s − 8.48i·11-s + 10.3·13-s − 21.2i·17-s − 20·19-s − 14.6i·23-s − 29·25-s − 36.7i·29-s − 51.9·31-s + 76.3i·35-s + 41.5·37-s + 72.1i·41-s + 40·43-s − 73.4i·47-s + ⋯ |
| L(s) = 1 | − 1.46i·5-s − 1.48·7-s − 0.771i·11-s + 0.799·13-s − 1.24i·17-s − 1.05·19-s − 0.638i·23-s − 1.15·25-s − 1.26i·29-s − 1.67·31-s + 2.18i·35-s + 1.12·37-s + 1.75i·41-s + 0.930·43-s − 1.56i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5838312627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5838312627\) |
| \(L(2)\) |
|
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 | \( 1 + 7.34iT - 25T^{2} \) |
| 7 | \( 1 + 10.3T + 49T^{2} \) |
| 11 | \( 1 + 8.48iT - 121T^{2} \) |
| 13 | \( 1 - 10.3T + 169T^{2} \) |
| 17 | \( 1 + 21.2iT - 289T^{2} \) |
| 19 | \( 1 + 20T + 361T^{2} \) |
| 23 | \( 1 + 14.6iT - 529T^{2} \) |
| 29 | \( 1 + 36.7iT - 841T^{2} \) |
| 31 | \( 1 + 51.9T + 961T^{2} \) |
| 37 | \( 1 - 41.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 72.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 40T + 1.84e3T^{2} \) |
| 47 | \( 1 + 73.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 36.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 33.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.72e3T^{2} \) |
| 67 | \( 1 + 100T + 4.48e3T^{2} \) |
| 71 | \( 1 + 73.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 20T + 5.32e3T^{2} \) |
| 79 | \( 1 - 51.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 127. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 12.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 40T + 9.40e3T^{2} \) |
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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
−8.484572850364494500426227134952, −7.71081379798575330030611226725, −6.51762177426436259539589015234, −6.05946304455884958806522186364, −5.16293701910768687090895482061, −4.23580639065801223972631129085, −3.45674327454576437847480472955, −2.36341095923150852238131432571, −0.874446351895503344352321540752, −0.17406383151243383918410930347,
1.74839264346501482224174085955, 2.80799604569383465595993447916, 3.55404649740844485069620985038, 4.19395666683100672097590801666, 5.93155442862215245174799442768, 6.09288373282015450709020734316, 7.10034933104836193753999043321, 7.40489781973582862997289246242, 8.721400181636294095761019571848, 9.348868545547980897639963466082
