| L(s) = 1 | + 1.12i·5-s − 7-s − 4.76i·11-s + 0.456i·13-s − 0.415·17-s + 7.63i·19-s + 1.58·23-s + 3.72·25-s − 6.72i·29-s + 5.89·31-s − 1.12i·35-s + 5.89i·37-s + 0.415·41-s − 9.43i·43-s + 11.2·47-s + ⋯ |
| L(s) = 1 | + 0.504i·5-s − 0.377·7-s − 1.43i·11-s + 0.126i·13-s − 0.100·17-s + 1.75i·19-s + 0.330·23-s + 0.745·25-s − 1.24i·29-s + 1.05·31-s − 0.190i·35-s + 0.969i·37-s + 0.0648·41-s − 1.43i·43-s + 1.64·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.650014974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.650014974\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 5 | \( 1 - 1.12iT - 5T^{2} \) |
| 11 | \( 1 + 4.76iT - 11T^{2} \) |
| 13 | \( 1 - 0.456iT - 13T^{2} \) |
| 17 | \( 1 + 0.415T + 17T^{2} \) |
| 19 | \( 1 - 7.63iT - 19T^{2} \) |
| 23 | \( 1 - 1.58T + 23T^{2} \) |
| 29 | \( 1 + 6.72iT - 29T^{2} \) |
| 31 | \( 1 - 5.89T + 31T^{2} \) |
| 37 | \( 1 - 5.89iT - 37T^{2} \) |
| 41 | \( 1 - 0.415T + 41T^{2} \) |
| 43 | \( 1 + 9.43iT - 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 7.63iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 1.80iT - 61T^{2} \) |
| 67 | \( 1 - 8.09iT - 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 3.34T + 73T^{2} \) |
| 79 | \( 1 - 4.83T + 79T^{2} \) |
| 83 | \( 1 - 5.53iT - 83T^{2} \) |
| 89 | \( 1 + 4.92T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−9.060297316618011360393908267875, −8.383244326301526406932086338756, −7.68893255201751616763650769989, −6.63396725537727447962734466408, −6.09970454820497602871831609113, −5.31265918490980503961840664810, −4.02302140691680418599375545910, −3.33331824525136858158142700221, −2.38436653837675331480194855860, −0.854943387791684233019425816412,
0.873376450381500187089627446108, 2.24344426066357262656921425869, 3.17737185504720546932259692444, 4.59479440538624739880361937631, 4.78647708173012603618860300187, 5.98894136767419643234945281007, 6.99786744355401008955885573196, 7.34388447148809585935172946547, 8.541630237300879767005009060131, 9.166963177113108615260778276429
