L(s) = 1 | + (−0.0695 + 0.120i)3-s − 3.40·7-s + (1.49 + 2.58i)9-s + 0.185·11-s + (−1.17 − 2.03i)13-s + (3.35 − 5.81i)17-s + (−2.14 + 3.79i)19-s + (0.236 − 0.410i)21-s + (−0.463 − 0.803i)23-s − 0.831·27-s + (0.677 + 1.17i)29-s − 9.46·31-s + (−0.0129 + 0.0223i)33-s − 1.62·37-s + 0.327·39-s + ⋯ |
L(s) = 1 | + (−0.0401 + 0.0695i)3-s − 1.28·7-s + (0.496 + 0.860i)9-s + 0.0559·11-s + (−0.326 − 0.565i)13-s + (0.814 − 1.41i)17-s + (−0.491 + 0.870i)19-s + (0.0516 − 0.0895i)21-s + (−0.0967 − 0.167i)23-s − 0.160·27-s + (0.125 + 0.217i)29-s − 1.69·31-s + (−0.00224 + 0.00389i)33-s − 0.267·37-s + 0.0524·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7480632697\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7480632697\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.14 - 3.79i)T \) |
good | 3 | \( 1 + (0.0695 - 0.120i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + 3.40T + 7T^{2} \) |
| 11 | \( 1 - 0.185T + 11T^{2} \) |
| 13 | \( 1 + (1.17 + 2.03i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.35 + 5.81i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (0.463 + 0.803i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.677 - 1.17i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 9.46T + 31T^{2} \) |
| 37 | \( 1 + 1.62T + 37T^{2} \) |
| 41 | \( 1 + (-4.29 + 7.44i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.80 + 6.58i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.09 - 7.09i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.70 + 11.6i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.02 + 3.51i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.31 + 10.9i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.62 + 2.81i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.59 + 11.4i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.45 + 4.24i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.98 - 5.16i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 1.30T + 83T^{2} \) |
| 89 | \( 1 + (-1.95 - 3.38i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.25 + 12.5i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.285968043376489066214094808667, −8.034272364946529634953724306542, −7.42287882703755486960524387165, −6.69338655503755617039921523483, −5.67811636064477076646831961690, −5.05421444811138936697814209496, −3.86408885779725159369698518323, −3.08122204026303428931669490698, −1.98525737789912676768419921692, −0.28308233325644593303004767071,
1.28474149138904556803795616018, 2.68486528679323345912179829383, 3.68548455800015398713136986168, 4.29330007586109223563378526123, 5.68193739994406799270147113966, 6.32448589314524394683700585190, 6.96430157400400236972485309199, 7.76396065023771415566691372855, 8.948967309468619612194950408651, 9.371633890641171161543838880268