| L(s) = 1 | + 2.60i·3-s − 2.76·7-s − 3.76·9-s − 2.16i·11-s + (−3.60 − 0.167i)13-s − 5.03i·19-s − 7.20i·21-s − 4.93i·23-s − 2.00i·27-s + 4.43·29-s − 3.37i·31-s + 5.63·33-s + 3.97·37-s + (0.434 − 9.37i)39-s − 1.66i·41-s + ⋯ |
| L(s) = 1 | + 1.50i·3-s − 1.04·7-s − 1.25·9-s − 0.653i·11-s + (−0.998 − 0.0463i)13-s − 1.15i·19-s − 1.57i·21-s − 1.02i·23-s − 0.384i·27-s + 0.823·29-s − 0.605i·31-s + 0.981·33-s + 0.653·37-s + (0.0695 − 1.50i)39-s − 0.260i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.405 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5190716677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5190716677\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (3.60 + 0.167i)T \) |
| good | 3 | \( 1 - 2.60iT - 3T^{2} \) |
| 7 | \( 1 + 2.76T + 7T^{2} \) |
| 11 | \( 1 + 2.16iT - 11T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 5.03iT - 19T^{2} \) |
| 23 | \( 1 + 4.93iT - 23T^{2} \) |
| 29 | \( 1 - 4.43T + 29T^{2} \) |
| 31 | \( 1 + 3.37iT - 31T^{2} \) |
| 37 | \( 1 - 3.97T + 37T^{2} \) |
| 41 | \( 1 + 1.66iT - 41T^{2} \) |
| 43 | \( 1 - 9.80iT - 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 12.7iT - 53T^{2} \) |
| 59 | \( 1 + 8.16iT - 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 7.10T + 67T^{2} \) |
| 71 | \( 1 - 16.1iT - 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 3.97T + 83T^{2} \) |
| 89 | \( 1 + 7.94iT - 89T^{2} \) |
| 97 | \( 1 - 0.462T + 97T^{2} \) |
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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
−9.816082761730066644616141173946, −8.941491726387750403649718608057, −8.126702582830332710231154494855, −6.85890874705967583728703332938, −6.13348015885346126439042264463, −5.00499929596449118904564439067, −4.45182463084605406932170030987, −3.32660745836864565218338221717, −2.68989525763417544475380019086, −0.21462817197938629799587231115,
1.39105679110539805771836358157, 2.43814277599021283746440751149, 3.44932128369986632656163607875, 4.82570427786468677879966646169, 5.98987217510176226359977236495, 6.56071483732148919028631805544, 7.42474048380335706938621152049, 7.79885782318120054292126158303, 8.965518175139884898069104292561, 9.785733491237096149063265219809
