| L(s) = 1 | + 3-s + 0.331·5-s + 3.08·7-s + 9-s − 3.69·11-s − 4.64·13-s + 0.331·15-s + 6.52·17-s − 0.867·19-s + 3.08·21-s + 4·23-s − 4.88·25-s + 27-s + 4.89·29-s + 6.14·31-s − 3.69·33-s + 1.02·35-s + 3.64·37-s − 4.64·39-s + 3.92·41-s + 3.92·43-s + 0.331·45-s − 1.65·47-s + 2.50·49-s + 6.52·51-s − 0.564·53-s − 1.22·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.148·5-s + 1.16·7-s + 0.333·9-s − 1.11·11-s − 1.28·13-s + 0.0856·15-s + 1.58·17-s − 0.198·19-s + 0.672·21-s + 0.834·23-s − 0.977·25-s + 0.192·27-s + 0.908·29-s + 1.10·31-s − 0.643·33-s + 0.172·35-s + 0.599·37-s − 0.743·39-s + 0.613·41-s + 0.599·43-s + 0.0494·45-s − 0.241·47-s + 0.357·49-s + 0.913·51-s − 0.0775·53-s − 0.165·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.624553668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.624553668\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 - 0.331T + 5T^{2} \) |
| 7 | \( 1 - 3.08T + 7T^{2} \) |
| 11 | \( 1 + 3.69T + 11T^{2} \) |
| 13 | \( 1 + 4.64T + 13T^{2} \) |
| 17 | \( 1 - 6.52T + 17T^{2} \) |
| 19 | \( 1 + 0.867T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 4.89T + 29T^{2} \) |
| 31 | \( 1 - 6.14T + 31T^{2} \) |
| 37 | \( 1 - 3.64T + 37T^{2} \) |
| 41 | \( 1 - 3.92T + 41T^{2} \) |
| 43 | \( 1 - 3.92T + 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 + 0.564T + 53T^{2} \) |
| 59 | \( 1 + 6.59T + 59T^{2} \) |
| 61 | \( 1 - 14.8T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 - 7.49T + 71T^{2} \) |
| 73 | \( 1 - 5.62T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 9.35T + 83T^{2} \) |
| 89 | \( 1 + 18.1T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−8.454956367021656275359362415460, −7.87437876966603384962219825889, −7.60897837041372955837497608647, −6.54003963551924492920373660410, −5.29778544656542004782161964159, −5.04161073636606118155229044021, −4.02775335820355838240834235375, −2.82953232918161496032334412599, −2.25749371706458260422115415852, −0.992522314747050444141214711465,
0.992522314747050444141214711465, 2.25749371706458260422115415852, 2.82953232918161496032334412599, 4.02775335820355838240834235375, 5.04161073636606118155229044021, 5.29778544656542004782161964159, 6.54003963551924492920373660410, 7.60897837041372955837497608647, 7.87437876966603384962219825889, 8.454956367021656275359362415460
