| L(s) = 1 | + 3-s + 2.41i·5-s − 4.14i·7-s + 9-s − 3.46i·11-s + 2.41i·15-s − 6.17·17-s − 3.46i·19-s − 4.14i·21-s − 2·23-s − 0.821·25-s + 27-s − 8.17·29-s − 7.60i·31-s − 3.46i·33-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.07i·5-s − 1.56i·7-s + 0.333·9-s − 1.04i·11-s + 0.622i·15-s − 1.49·17-s − 0.794i·19-s − 0.904i·21-s − 0.417·23-s − 0.164·25-s + 0.192·27-s − 1.51·29-s − 1.36i·31-s − 0.603i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.392309746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.392309746\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2.41iT - 5T^{2} \) |
| 7 | \( 1 + 4.14iT - 7T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 17 | \( 1 + 6.17T + 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 8.17T + 29T^{2} \) |
| 31 | \( 1 + 7.60iT - 31T^{2} \) |
| 37 | \( 1 + 1.05iT - 37T^{2} \) |
| 41 | \( 1 + 5.87iT - 41T^{2} \) |
| 43 | \( 1 + 0.821T + 43T^{2} \) |
| 47 | \( 1 - 10.3iT - 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 1.36iT - 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 + 4.14iT - 67T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 - 11.3iT - 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 11.7iT - 83T^{2} \) |
| 89 | \( 1 + 6.92iT - 89T^{2} \) |
| 97 | \( 1 + 2.04iT - 97T^{2} \) |
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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
−8.973985124439551345012596508736, −7.963497689850963959228959824218, −7.34377852197147506140876813142, −6.73426860761537748078573438499, −5.98350820134608723701914120711, −4.55269946029445993331481679727, −3.83893181876032245036795970551, −3.08669180363082842319754831374, −2.04898218860139627871384793502, −0.42646246076505322766501673379,
1.71665635861250592616945937889, 2.28241861769894587451993170951, 3.57261184775461526262800023584, 4.67160039215686509547828211080, 5.18326476424364760916908248049, 6.16857140018869386227928497037, 7.07212239913188829222512767924, 8.165576706206760604630601507491, 8.602987137998186593987112795019, 9.287244811273977984775908870030
