| L(s) = 1 | + 2.73i·3-s − 2i·7-s − 4.46·9-s − 3.46·11-s − 2.73i·13-s + 3.46i·17-s − 19-s + 5.46·21-s − 3.46i·23-s − 3.99i·27-s − 3.46·29-s − 1.46·31-s − 9.46i·33-s − 6.73i·37-s + 7.46·39-s + ⋯ |
| L(s) = 1 | + 1.57i·3-s − 0.755i·7-s − 1.48·9-s − 1.04·11-s − 0.757i·13-s + 0.840i·17-s − 0.229·19-s + 1.19·21-s − 0.722i·23-s − 0.769i·27-s − 0.643·29-s − 0.262·31-s − 1.64i·33-s − 1.10i·37-s + 1.19·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6613437874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6613437874\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.73iT - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 2.73iT - 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 + 6.73iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4.92iT - 43T^{2} \) |
| 47 | \( 1 + 12.9iT - 47T^{2} \) |
| 53 | \( 1 + 10.7iT - 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 6.73iT - 67T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 + 0.535iT - 73T^{2} \) |
| 79 | \( 1 + 2.92T + 79T^{2} \) |
| 83 | \( 1 - 3.46iT - 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 16.5iT - 97T^{2} \) |
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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
−9.162139958254118214005248897832, −8.414523565307635688484526316401, −7.69998132180816987453692019551, −6.65122693251984489817794257360, −5.45895404721001998588141621011, −5.07764536967559769974554854520, −3.94757589520230145568180429970, −3.54608357388191831612022758817, −2.28078474052169429511839132782, −0.23667438150447221158205953482,
1.35632934320536687730304894410, 2.32410381523485315122514870649, 3.04697598017794262047488864882, 4.62423123865531631987689892022, 5.58943526163631367155100796194, 6.22584884157680453077185216869, 7.14411658315026498063411798209, 7.63185697457803680906269437462, 8.421033077023396698866351859550, 9.151553390492859181763123869064
