| L(s) = 1 | + 3.86i·5-s + i·7-s + 5.27·11-s − 2·13-s − 1.03i·17-s + 5.46i·19-s − 0.378·23-s − 9.92·25-s + 6.31i·29-s + 9.46i·31-s − 3.86·35-s + 10.9·37-s − 5.93i·41-s − 4i·43-s − 8.48·47-s + ⋯ |
| L(s) = 1 | + 1.72i·5-s + 0.377i·7-s + 1.59·11-s − 0.554·13-s − 0.251i·17-s + 1.25i·19-s − 0.0790·23-s − 1.98·25-s + 1.17i·29-s + 1.69i·31-s − 0.653·35-s + 1.79·37-s − 0.926i·41-s − 0.609i·43-s − 1.23·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.742528316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.742528316\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| good | 5 | \( 1 - 3.86iT - 5T^{2} \) |
| 11 | \( 1 - 5.27T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 1.03iT - 17T^{2} \) |
| 19 | \( 1 - 5.46iT - 19T^{2} \) |
| 23 | \( 1 + 0.378T + 23T^{2} \) |
| 29 | \( 1 - 6.31iT - 29T^{2} \) |
| 31 | \( 1 - 9.46iT - 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 5.93iT - 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 + 7.07iT - 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 - 3.46T + 61T^{2} \) |
| 67 | \( 1 - 15.4iT - 67T^{2} \) |
| 71 | \( 1 - 4.52T + 71T^{2} \) |
| 73 | \( 1 - 0.535T + 73T^{2} \) |
| 79 | \( 1 + 10.3iT - 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 - 3.86iT - 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.779485404245481533560463814592, −7.88107137417574315273135699295, −7.06416439647861408376558128955, −6.64439621502051578554713340597, −6.01810714068244615496734739015, −5.06976480547029620900652547161, −3.86830569461146443681549584490, −3.37581576242491460827708332367, −2.46499565682000494933980818291, −1.48905002221123286265998450509,
0.53242413912043393420610075780, 1.30049939429184562092535839994, 2.40588076424096118894316504360, 3.81873395268298226979084717754, 4.45553993697964300048250802092, 4.87671871012733308696104221099, 6.02961771761026726321634392459, 6.49565197298281143857832476079, 7.69394769421503083317956805111, 8.075535939437613863130566458508
