| L(s) = 1 | − 0.638i·3-s + 9.06·7-s + 8.59·9-s + (10.0 − 4.36i)11-s + 10.4·13-s − 3.39·17-s + 28.0i·19-s − 5.79i·21-s + 18.8i·23-s − 11.2i·27-s − 22.2i·29-s + 0.00744·31-s + (−2.79 − 6.44i)33-s − 8.96i·37-s − 6.66i·39-s + ⋯ |
| L(s) = 1 | − 0.212i·3-s + 1.29·7-s + 0.954·9-s + (0.917 − 0.397i)11-s + 0.802·13-s − 0.199·17-s + 1.47i·19-s − 0.275i·21-s + 0.818i·23-s − 0.416i·27-s − 0.766i·29-s + 0.000240·31-s + (−0.0845 − 0.195i)33-s − 0.242i·37-s − 0.170i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0552i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.748753438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.748753438\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-10.0 + 4.36i)T \) |
| good | 3 | \( 1 + 0.638iT - 9T^{2} \) |
| 7 | \( 1 - 9.06T + 49T^{2} \) |
| 13 | \( 1 - 10.4T + 169T^{2} \) |
| 17 | \( 1 + 3.39T + 289T^{2} \) |
| 19 | \( 1 - 28.0iT - 361T^{2} \) |
| 23 | \( 1 - 18.8iT - 529T^{2} \) |
| 29 | \( 1 + 22.2iT - 841T^{2} \) |
| 31 | \( 1 - 0.00744T + 961T^{2} \) |
| 37 | \( 1 + 8.96iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 16.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 81.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 59.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 60.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 81.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 114. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 99.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 70.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 16.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 118. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 30.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 4.02T + 7.92e3T^{2} \) |
| 97 | \( 1 + 101. iT - 9.40e3T^{2} \) |
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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
−9.724206684402755841634720014921, −8.629174212473010662257713252996, −8.086284056728099113495474586310, −7.22565541827773303481637569476, −6.29268392642838882268243830391, −5.40951930372990107703018270758, −4.28768759817967898958307307488, −3.62743282526748732938144651201, −1.84309672788405366526696531289, −1.21329306751475330604916972015,
1.09128666757457280968411202562, 2.03872289205678988188560230510, 3.59644924325068676063598104385, 4.57076768097940093162492120391, 5.05332551760743400284251069635, 6.52703573728064011714095171812, 7.04683073707715862643865968576, 8.149411152527904032720221561303, 8.815663197003399790053356608135, 9.622938704573696591495389525009
