| L(s) = 1 | + 2.49i·3-s + 2.49i·7-s − 3.20·9-s − 3.49·11-s + 0.713i·13-s + 3.91i·17-s + 19-s − 6.20·21-s + 2.20i·23-s − 0.509i·27-s − 0.636·29-s − 2.14·31-s − 8.69i·33-s + 2.06i·37-s − 1.77·39-s + ⋯ |
| L(s) = 1 | + 1.43i·3-s + 0.941i·7-s − 1.06·9-s − 1.05·11-s + 0.197i·13-s + 0.950i·17-s + 0.229·19-s − 1.35·21-s + 0.459i·23-s − 0.0979i·27-s − 0.118·29-s − 0.384·31-s − 1.51i·33-s + 0.339i·37-s − 0.284·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7776367197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7776367197\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.49iT - 3T^{2} \) |
| 7 | \( 1 - 2.49iT - 7T^{2} \) |
| 11 | \( 1 + 3.49T + 11T^{2} \) |
| 13 | \( 1 - 0.713iT - 13T^{2} \) |
| 17 | \( 1 - 3.91iT - 17T^{2} \) |
| 23 | \( 1 - 2.20iT - 23T^{2} \) |
| 29 | \( 1 + 0.636T + 29T^{2} \) |
| 31 | \( 1 + 2.14T + 31T^{2} \) |
| 37 | \( 1 - 2.06iT - 37T^{2} \) |
| 41 | \( 1 + 4.35T + 41T^{2} \) |
| 43 | \( 1 - 4.55iT - 43T^{2} \) |
| 47 | \( 1 - 0.268iT - 47T^{2} \) |
| 53 | \( 1 + 7.98iT - 53T^{2} \) |
| 59 | \( 1 - 2.57T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 1.08iT - 67T^{2} \) |
| 71 | \( 1 - 6.83T + 71T^{2} \) |
| 73 | \( 1 + 15.0iT - 73T^{2} \) |
| 79 | \( 1 - 7.63T + 79T^{2} \) |
| 83 | \( 1 + 0.923iT - 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 - 6.14iT - 97T^{2} \) |
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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
−9.161579232077970635865191661558, −8.374676896734722441570468794020, −7.79288666685536375576513616230, −6.63980853911640696104331333460, −5.70059397161161435704024811313, −5.24321224224846928409430435005, −4.50899746438590120917984078089, −3.60662581388110246341105251411, −2.90061353124348997683568267497, −1.83616106002060038410262931476,
0.23519039623746656479886125353, 1.14088560416699761886137482040, 2.25296497467054431748179219903, 3.00900683175834972874577745819, 4.14169948379037910479312859877, 5.13800651578231756219686026568, 5.85159655395787988159237126191, 6.86530907499389929677450502919, 7.21125607724595925161636679169, 7.82257974461504780548865685208
