| L(s) = 1 | − 2i·3-s + (5 − 10i)5-s − 26i·7-s + 23·9-s + 28·11-s − 12i·13-s + (−20 − 10i)15-s + 64i·17-s − 60·19-s − 52·21-s − 58i·23-s + (−75 − 100i)25-s − 100i·27-s + 90·29-s − 128·31-s + ⋯ |
| L(s) = 1 | − 0.384i·3-s + (0.447 − 0.894i)5-s − 1.40i·7-s + 0.851·9-s + 0.767·11-s − 0.256i·13-s + (−0.344 − 0.172i)15-s + 0.913i·17-s − 0.724·19-s − 0.540·21-s − 0.525i·23-s + (−0.599 − 0.800i)25-s − 0.712i·27-s + 0.576·29-s − 0.741·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.037591275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.037591275\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-5 + 10i)T \) |
| good | 3 | \( 1 + 2iT - 27T^{2} \) |
| 7 | \( 1 + 26iT - 343T^{2} \) |
| 11 | \( 1 - 28T + 1.33e3T^{2} \) |
| 13 | \( 1 + 12iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 64iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 60T + 6.85e3T^{2} \) |
| 23 | \( 1 + 58iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 90T + 2.43e4T^{2} \) |
| 31 | \( 1 + 128T + 2.97e4T^{2} \) |
| 37 | \( 1 - 236iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 242T + 6.89e4T^{2} \) |
| 43 | \( 1 + 362iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 226iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 108iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 20T + 2.05e5T^{2} \) |
| 61 | \( 1 + 542T + 2.26e5T^{2} \) |
| 67 | \( 1 + 434iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 632iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 720T + 4.93e5T^{2} \) |
| 83 | \( 1 - 478iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 490T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.45e3iT - 9.12e5T^{2} \) |
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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
−10.65293564668183674781522240351, −10.11556071257465796326139075582, −8.995261770823402169029000634345, −8.010867569235655360101214641531, −7.00129800582575677014624815039, −6.13574437147212438980891751340, −4.61104182251339335596954819002, −3.89482963891029657179100245135, −1.75232921695408700796812412207, −0.77660257682822605928917914978,
1.84391091884897757701568395479, 3.02336676388724617027351559708, 4.39864792936923166987060598602, 5.68297296987992963923648045740, 6.55304683624049432613562631405, 7.56716214727069016926041410952, 9.119870743668240181451128152218, 9.436179251195188422269726643836, 10.57414412333405307026209586741, 11.44882135083684824148929030341
