| 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
−11.44882135083684824148929030341, −10.57414412333405307026209586741, −9.436179251195188422269726643836, −9.119870743668240181451128152218, −7.56716214727069016926041410952, −6.55304683624049432613562631405, −5.68297296987992963923648045740, −4.39864792936923166987060598602, −3.02336676388724617027351559708, −1.84391091884897757701568395479,
0.77660257682822605928917914978, 1.75232921695408700796812412207, 3.89482963891029657179100245135, 4.61104182251339335596954819002, 6.13574437147212438980891751340, 7.00129800582575677014624815039, 8.010867569235655360101214641531, 8.995261770823402169029000634345, 10.11556071257465796326139075582, 10.65293564668183674781522240351
