| L(s) = 1 | + 2.64i·7-s − 5.29i·11-s − 5.29i·23-s + 5·25-s − 2·29-s − 6·37-s + 5.29i·43-s − 7.00·49-s − 10·53-s − 15.8i·67-s − 5.29i·71-s + 14.0·77-s − 15.8i·79-s − 5.29i·107-s + 18·109-s + ⋯ |
| L(s) = 1 | + 0.999i·7-s − 1.59i·11-s − 1.10i·23-s + 25-s − 0.371·29-s − 0.986·37-s + 0.806i·43-s − 49-s − 1.37·53-s − 1.93i·67-s − 0.627i·71-s + 1.59·77-s − 1.78i·79-s − 0.511i·107-s + 1.72·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.283810032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.283810032\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 + 5.29iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 5.29iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 5.29iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 15.8iT - 67T^{2} \) |
| 71 | \( 1 + 5.29iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 15.8iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.411715803373460716983072648287, −7.68810055061322622935243639120, −6.52701038884928792590406135849, −6.14612933323413287775395457128, −5.30617935676239931266497629420, −4.61390478063896448842881982765, −3.36588401064525294584337415258, −2.88198313905676551098254086385, −1.74118179592128553306189160726, −0.38360277681249166375749790802,
1.19128981666386702921935889814, 2.10629265148899205655608464761, 3.29792091092080667278049004127, 4.10490703329801993545648018676, 4.80396695372781443002574279281, 5.54077655308713678843217833674, 6.72757210428033329034350247874, 7.13279832378630315776845523664, 7.70904374231593601463597270838, 8.609574947517546075962265079641
