| L(s) = 1 | + 1.41·2-s − 1.73i·3-s + 2.00·4-s − 2.44i·6-s + (−6.73 − 1.91i)7-s + 2.82·8-s − 2.99·9-s + 17.5·11-s − 3.46i·12-s − 4.83i·13-s + (−9.52 − 2.70i)14-s + 4.00·16-s − 18.0i·17-s − 4.24·18-s + 9.13i·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577i·3-s + 0.500·4-s − 0.408i·6-s + (−0.961 − 0.273i)7-s + 0.353·8-s − 0.333·9-s + 1.59·11-s − 0.288i·12-s − 0.371i·13-s + (−0.680 − 0.193i)14-s + 0.250·16-s − 1.06i·17-s − 0.235·18-s + 0.480i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.273 + 0.961i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.273 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.494102707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.494102707\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (6.73 + 1.91i)T \) |
| good | 11 | \( 1 - 17.5T + 121T^{2} \) |
| 13 | \( 1 + 4.83iT - 169T^{2} \) |
| 17 | \( 1 + 18.0iT - 289T^{2} \) |
| 19 | \( 1 - 9.13iT - 361T^{2} \) |
| 23 | \( 1 - 3.72T + 529T^{2} \) |
| 29 | \( 1 + 1.12T + 841T^{2} \) |
| 31 | \( 1 + 57.0iT - 961T^{2} \) |
| 37 | \( 1 + 41.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 11.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 64.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 77.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 77.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 87.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 5.36iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 47.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 58.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 53.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 74.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 28.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 107. iT - 9.40e3T^{2} \) |
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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
−9.527059195775294630457900732835, −8.659768792715654191738900748738, −7.48216978292959408506455322517, −6.80103334622485728190893853014, −6.20518940346940518387531735404, −5.23784885399696381270305129519, −3.95681615584035583552891442849, −3.29064831478145825659680455824, −2.01122541492158013591872560269, −0.60756581594859515734202599802,
1.51728243168081129415651651456, 3.02622337768977804958369538044, 3.76188579124059254208193032233, 4.58219644475041337064914093386, 5.70738662177824974976982830232, 6.50136269816931885576060772731, 7.05684435127151230274887030114, 8.623226439659998116791580213221, 9.080085241273402089747812742658, 10.04990580631707803773510497509
