| L(s) = 1 | + 1.73·3-s − 2.23i·5-s − 1.69i·7-s − 5.98·9-s − 1.15i·11-s + 16.7·13-s − 3.87i·15-s + 4.61i·17-s + 1.53i·19-s − 2.94i·21-s + (21.7 − 7.39i)23-s − 5.00·25-s − 26.0·27-s + 16.0·29-s + 4.40·31-s + ⋯ |
| L(s) = 1 | + 0.578·3-s − 0.447i·5-s − 0.242i·7-s − 0.665·9-s − 0.105i·11-s + 1.29·13-s − 0.258i·15-s + 0.271i·17-s + 0.0806i·19-s − 0.140i·21-s + (0.946 − 0.321i)23-s − 0.200·25-s − 0.963·27-s + 0.553·29-s + 0.142·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.279355817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.279355817\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-21.7 + 7.39i)T \) |
| good | 3 | \( 1 - 1.73T + 9T^{2} \) |
| 7 | \( 1 + 1.69iT - 49T^{2} \) |
| 11 | \( 1 + 1.15iT - 121T^{2} \) |
| 13 | \( 1 - 16.7T + 169T^{2} \) |
| 17 | \( 1 - 4.61iT - 289T^{2} \) |
| 19 | \( 1 - 1.53iT - 361T^{2} \) |
| 29 | \( 1 - 16.0T + 841T^{2} \) |
| 31 | \( 1 - 4.40T + 961T^{2} \) |
| 37 | \( 1 + 42.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 1.09T + 1.68e3T^{2} \) |
| 43 | \( 1 - 44.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 36.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 67.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 71.9T + 3.48e3T^{2} \) |
| 61 | \( 1 + 98.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 62.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 59.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 67.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 58.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 151. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 62.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 192. iT - 9.40e3T^{2} \) |
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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
−8.703338020458108591836837287624, −8.395723212885679923098043846784, −7.50043115170779185087187814965, −6.45341798925850099763318450775, −5.74837017967483495677281245340, −4.76746675850219703495592113681, −3.75694685214964009714574652288, −3.03161583532207866396749388115, −1.81246652711458879171705275454, −0.60371035908226709675341309564,
1.14287006044486369176041487370, 2.49623349019969517336501920272, 3.17951709193254186199099762833, 4.06780530770912057133803514462, 5.27721967872855177206980145090, 6.04358182389877626082050280919, 6.87800208683664428063681842874, 7.72395748506611874538024522580, 8.698082311715478428051737665526, 8.873413737199897473016006283674
