| L(s) = 1 | − 1.73i·3-s − 9.43i·7-s − 2.99·9-s + 3.73i·11-s − 12.5·13-s − 30.0·17-s − 10.7i·19-s − 16.3·21-s + 38.0i·23-s + 5.19i·27-s + 27.0·29-s − 16.9i·31-s + 6.46·33-s − 45.2·37-s + 21.8i·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.34i·7-s − 0.333·9-s + 0.339i·11-s − 0.968·13-s − 1.76·17-s − 0.565i·19-s − 0.778·21-s + 1.65i·23-s + 0.192i·27-s + 0.931·29-s − 0.547i·31-s + 0.195·33-s − 1.22·37-s + 0.559i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8989996653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8989996653\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 9.43iT - 49T^{2} \) |
| 11 | \( 1 - 3.73iT - 121T^{2} \) |
| 13 | \( 1 + 12.5T + 169T^{2} \) |
| 17 | \( 1 + 30.0T + 289T^{2} \) |
| 19 | \( 1 + 10.7iT - 361T^{2} \) |
| 23 | \( 1 - 38.0iT - 529T^{2} \) |
| 29 | \( 1 - 27.0T + 841T^{2} \) |
| 31 | \( 1 + 16.9iT - 961T^{2} \) |
| 37 | \( 1 + 45.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 68.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 81.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 17.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 50.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 63.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 12.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 10.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 1.92iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 31.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 69.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 67.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 63.1T + 7.92e3T^{2} \) |
| 97 | \( 1 + 51.3T + 9.40e3T^{2} \) |
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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
−8.912613491406442456911996542801, −7.86190298109598316292972995518, −7.25848486623489818853844473148, −6.85084299934662588726669033430, −5.88882830326822261780329789854, −4.71144365912925683479343573557, −4.23244211634690270149812422137, −3.00199176319657712842487562739, −2.01705082680750097898195756376, −0.881526949983670672544354644899,
0.25448686168688357542028227468, 2.20346773090912233485478405211, 2.63956542551117550379023003976, 3.90966036483437793433398861232, 4.79499711484198366764015965672, 5.42024241878984336813617313163, 6.33312918009783400320395144386, 6.99160705413761502506214812575, 8.274968872417808594457528801817, 8.744252899862948634529243075205
