| L(s) = 1 | + (−1.52 + 1.63i)5-s + (0.114 + 2.64i)7-s − 3.82i·11-s + 6.36·13-s + 0.444i·17-s + 4.39i·19-s + 4.65·23-s + (−0.345 − 4.98i)25-s − 9.53i·29-s − 4.88i·31-s + (−4.49 − 3.84i)35-s + 5.61i·37-s + 8.01·41-s + 3.33i·43-s − 12.8i·47-s + ⋯ |
| L(s) = 1 | + (−0.682 + 0.731i)5-s + (0.0433 + 0.999i)7-s − 1.15i·11-s + 1.76·13-s + 0.107i·17-s + 1.00i·19-s + 0.971·23-s + (−0.0691 − 0.997i)25-s − 1.77i·29-s − 0.877i·31-s + (−0.760 − 0.649i)35-s + 0.922i·37-s + 1.25·41-s + 0.508i·43-s − 1.87i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.864425697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.864425697\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.52 - 1.63i)T \) |
| 7 | \( 1 + (-0.114 - 2.64i)T \) |
| good | 11 | \( 1 + 3.82iT - 11T^{2} \) |
| 13 | \( 1 - 6.36T + 13T^{2} \) |
| 17 | \( 1 - 0.444iT - 17T^{2} \) |
| 19 | \( 1 - 4.39iT - 19T^{2} \) |
| 23 | \( 1 - 4.65T + 23T^{2} \) |
| 29 | \( 1 + 9.53iT - 29T^{2} \) |
| 31 | \( 1 + 4.88iT - 31T^{2} \) |
| 37 | \( 1 - 5.61iT - 37T^{2} \) |
| 41 | \( 1 - 8.01T + 41T^{2} \) |
| 43 | \( 1 - 3.33iT - 43T^{2} \) |
| 47 | \( 1 + 12.8iT - 47T^{2} \) |
| 53 | \( 1 + 1.27T + 53T^{2} \) |
| 59 | \( 1 + 5.81T + 59T^{2} \) |
| 61 | \( 1 + 5.28iT - 61T^{2} \) |
| 67 | \( 1 + 9.42iT - 67T^{2} \) |
| 71 | \( 1 - 2.35iT - 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 - 1.09iT - 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 4.80T + 97T^{2} \) |
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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
−8.181607983550302923889120096455, −7.84538256874883677904821247547, −6.60831985458540822114903104184, −6.08294956618784502833323834509, −5.66417455184124523056745841914, −4.41625889388667263747907331596, −3.57267734091248987115321879198, −3.09543501437546461053659174396, −2.03739384529374140035036563305, −0.71348013386630768050685611485,
0.872986592141273313654984831451, 1.50825529027533471279048726583, 3.03162537531602721445836805651, 3.81133103256664382416735806393, 4.51831779649062973968029677856, 5.03262523144704916756061254246, 6.08161884796664416368187783767, 7.16774945754712999735183991164, 7.22482571572903849245136528857, 8.232150312363895344663300674628
