| L(s) = 1 | + 12i·7-s − 5.65i·11-s + 8i·13-s − 9.89·17-s + 16·19-s − 39.5·23-s + 29.6i·29-s − 4·31-s + 30i·37-s − 21.2i·41-s + 8i·43-s + 16.9·47-s − 95·49-s + 49.4·53-s − 79.1i·59-s + ⋯ |
| L(s) = 1 | + 1.71i·7-s − 0.514i·11-s + 0.615i·13-s − 0.582·17-s + 0.842·19-s − 1.72·23-s + 1.02i·29-s − 0.129·31-s + 0.810i·37-s − 0.517i·41-s + 0.186i·43-s + 0.361·47-s − 1.93·49-s + 0.933·53-s − 1.34i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5794139330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5794139330\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 12iT - 49T^{2} \) |
| 11 | \( 1 + 5.65iT - 121T^{2} \) |
| 13 | \( 1 - 8iT - 169T^{2} \) |
| 17 | \( 1 + 9.89T + 289T^{2} \) |
| 19 | \( 1 - 16T + 361T^{2} \) |
| 23 | \( 1 + 39.5T + 529T^{2} \) |
| 29 | \( 1 - 29.6iT - 841T^{2} \) |
| 31 | \( 1 + 4T + 961T^{2} \) |
| 37 | \( 1 - 30iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 21.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 16.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 49.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 79.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 14T + 3.72e3T^{2} \) |
| 67 | \( 1 + 88iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 28.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 80iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 100T + 6.24e3T^{2} \) |
| 83 | \( 1 + 130.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 148. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 112iT - 9.40e3T^{2} \) |
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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
−9.363984141778170820953539083207, −8.721542670985086120609905982458, −8.168072206269239062872997782182, −7.07207350396058704370460248443, −6.14009263723160638763347959548, −5.59815722542855644557719042901, −4.70674480988396485014163262800, −3.53766173370996745488686584429, −2.56067438868451252233752527692, −1.67413134367032139399551538030,
0.15059105342851282699049540637, 1.27648315221602330582820402923, 2.57471257507828059687086241614, 3.90430961237801907542236904428, 4.24893708065022271255664420745, 5.42813105171589147982448287431, 6.35044507452718468565454183081, 7.33598851364626529908662212187, 7.63655958991451434598557329615, 8.585205836025527304379369177952
