| L(s) = 1 | + 0.585i·5-s − 1.41·7-s + 0.828i·11-s + 4.82i·13-s + 0.828·17-s − 2.82i·19-s + 6.82·23-s + 4.65·25-s − 4.58i·29-s − 7.07·31-s − 0.828i·35-s − 0.343i·37-s + 6.48·41-s − 1.17i·43-s + 4.48·47-s + ⋯ |
| L(s) = 1 | + 0.261i·5-s − 0.534·7-s + 0.249i·11-s + 1.33i·13-s + 0.200·17-s − 0.648i·19-s + 1.42·23-s + 0.931·25-s − 0.851i·29-s − 1.27·31-s − 0.140i·35-s − 0.0564i·37-s + 1.01·41-s − 0.178i·43-s + 0.654·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.465187906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.465187906\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.585iT - 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 - 0.828iT - 11T^{2} \) |
| 13 | \( 1 - 4.82iT - 13T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 19 | \( 1 + 2.82iT - 19T^{2} \) |
| 23 | \( 1 - 6.82T + 23T^{2} \) |
| 29 | \( 1 + 4.58iT - 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 + 0.343iT - 37T^{2} \) |
| 41 | \( 1 - 6.48T + 41T^{2} \) |
| 43 | \( 1 + 1.17iT - 43T^{2} \) |
| 47 | \( 1 - 4.48T + 47T^{2} \) |
| 53 | \( 1 - 10.2iT - 53T^{2} \) |
| 59 | \( 1 + 9.65iT - 59T^{2} \) |
| 61 | \( 1 - 11.6iT - 61T^{2} \) |
| 67 | \( 1 - 5.65iT - 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 3.17iT - 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−8.770530263425298741895693318876, −7.52006931607352854164824784310, −7.06948528109066579612778786162, −6.45878100855130823707951267950, −5.61908574372158512882558924097, −4.70655800201730804261418256633, −4.03263756994465670584595030925, −3.04808974059453861768476459113, −2.29703732740424848151563152689, −1.07800865279518924804938854984,
0.46060787125554904653390796048, 1.53172626243932995646848647906, 2.97061347047145867941358890534, 3.30130091540900443964038121269, 4.43031995819231888655314702903, 5.36921513328922206585820741822, 5.75252760690449249269375083566, 6.79279070973446012334692514131, 7.36310953356364727895870971359, 8.211048078236173698719940155855
