| L(s) = 1 | − 1.41·2-s + 1.73i·3-s + 2.00·4-s − 2.44i·6-s + (−3.84 + 5.85i)7-s − 2.82·8-s − 2.99·9-s + 20.2·11-s + 3.46i·12-s − 11.5i·13-s + (5.43 − 8.27i)14-s + 4.00·16-s − 26.7i·17-s + 4.24·18-s + 9.76i·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577i·3-s + 0.500·4-s − 0.408i·6-s + (−0.548 + 0.835i)7-s − 0.353·8-s − 0.333·9-s + 1.84·11-s + 0.288i·12-s − 0.889i·13-s + (0.388 − 0.591i)14-s + 0.250·16-s − 1.57i·17-s + 0.235·18-s + 0.513i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.548i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.835 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.091746511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.091746511\) |
| \(L(2)\) |
|
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 | \( 1 + (3.84 - 5.85i)T \) |
| good | 11 | \( 1 - 20.2T + 121T^{2} \) |
| 13 | \( 1 + 11.5iT - 169T^{2} \) |
| 17 | \( 1 + 26.7iT - 289T^{2} \) |
| 19 | \( 1 - 9.76iT - 361T^{2} \) |
| 23 | \( 1 + 34.8T + 529T^{2} \) |
| 29 | \( 1 + 10.4T + 841T^{2} \) |
| 31 | \( 1 + 39.3iT - 961T^{2} \) |
| 37 | \( 1 - 11.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 49.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 10.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 34.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 68.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 49.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 47.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 111.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 132.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 130. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 3.57T + 6.24e3T^{2} \) |
| 83 | \( 1 - 101. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.58iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 154. iT - 9.40e3T^{2} \) |
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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
−9.475232189118683933713065997430, −9.132632856316619796481409271751, −8.174393530164515871773292480248, −7.21542138420512976486789866022, −6.16807627855262424873176322092, −5.61702249326393739729473576341, −4.20078998428404277604949565509, −3.26029078885334488283785501499, −2.12181624648727018894897611351, −0.49673226279298219007116623888,
1.08234957674452935686043440528, 1.93135813650626187030593967487, 3.56671519772549718824177563948, 4.24668751556449061384689817229, 6.04648753747806663120267445261, 6.58089667771843654790698470212, 7.17358651546301060587429516100, 8.235495211889030334581219504565, 8.945414425060744123750113970240, 9.719974052288243629884683239492
