| L(s) = 1 | − 1.73i·3-s + 4·5-s + 10·7-s − 2.99·9-s − 10·11-s − 24.2i·13-s − 6.92i·15-s + 10·17-s − 19·19-s − 17.3i·21-s + 20·23-s − 9·25-s + 5.19i·27-s − 34.6i·29-s − 17.3i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.800·5-s + 1.42·7-s − 0.333·9-s − 0.909·11-s − 1.86i·13-s − 0.461i·15-s + 0.588·17-s − 19-s − 0.824i·21-s + 0.869·23-s − 0.359·25-s + 0.192i·27-s − 1.19i·29-s − 0.558i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.275779907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.275779907\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 19 | \( 1 + 19T \) |
| good | 5 | \( 1 - 4T + 25T^{2} \) |
| 7 | \( 1 - 10T + 49T^{2} \) |
| 11 | \( 1 + 10T + 121T^{2} \) |
| 13 | \( 1 + 24.2iT - 169T^{2} \) |
| 17 | \( 1 - 10T + 289T^{2} \) |
| 23 | \( 1 - 20T + 529T^{2} \) |
| 29 | \( 1 + 34.6iT - 841T^{2} \) |
| 31 | \( 1 + 17.3iT - 961T^{2} \) |
| 37 | \( 1 + 10.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 34.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 10T + 1.84e3T^{2} \) |
| 47 | \( 1 - 80T + 2.20e3T^{2} \) |
| 53 | \( 1 - 41.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 34.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 10T + 3.72e3T^{2} \) |
| 67 | \( 1 - 76.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 103. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 10T + 5.32e3T^{2} \) |
| 79 | \( 1 - 17.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 70T + 6.88e3T^{2} \) |
| 89 | \( 1 + 103. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 76.2iT - 9.40e3T^{2} \) |
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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
−9.825955110803486473240860427349, −8.584515737250789943519832300435, −7.941212978899379177460443938291, −7.42975830915162017074287945280, −5.91085941546642095568469417666, −5.54961374791959487081050150798, −4.52278658674846324482867718016, −2.87972752214784247179963544364, −2.00444441928150031958455513146, −0.74183601483777239773206508222,
1.54978647183617608738700598063, 2.43605058558670049919778434169, 4.00032308473241694158960517370, 4.89512405196761359331395626906, 5.49543040289886543915156586544, 6.66507306117713345129971012041, 7.62540078567229978575648815044, 8.688639601444591282864973441124, 9.132228358088393821893608003402, 10.24259394236443046104233393133
