| L(s) = 1 | + (1.06 − 1.96i)5-s − 3.96i·7-s − 0.632·11-s + 2.50i·13-s + 5.58i·17-s − 5.75·19-s − i·23-s + (−2.72 − 4.19i)25-s − 4.05·29-s − 0.852·31-s + (−7.78 − 4.22i)35-s + 8.51i·37-s − 6.20·41-s − 7.38i·43-s − 5.59i·47-s + ⋯ |
| L(s) = 1 | + (0.477 − 0.878i)5-s − 1.49i·7-s − 0.190·11-s + 0.695i·13-s + 1.35i·17-s − 1.32·19-s − 0.208i·23-s + (−0.544 − 0.838i)25-s − 0.753·29-s − 0.153·31-s + (−1.31 − 0.714i)35-s + 1.39i·37-s − 0.969·41-s − 1.12i·43-s − 0.815i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 - 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3742244694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3742244694\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.06 + 1.96i)T \) |
| 23 | \( 1 + iT \) |
| good | 7 | \( 1 + 3.96iT - 7T^{2} \) |
| 11 | \( 1 + 0.632T + 11T^{2} \) |
| 13 | \( 1 - 2.50iT - 13T^{2} \) |
| 17 | \( 1 - 5.58iT - 17T^{2} \) |
| 19 | \( 1 + 5.75T + 19T^{2} \) |
| 29 | \( 1 + 4.05T + 29T^{2} \) |
| 31 | \( 1 + 0.852T + 31T^{2} \) |
| 37 | \( 1 - 8.51iT - 37T^{2} \) |
| 41 | \( 1 + 6.20T + 41T^{2} \) |
| 43 | \( 1 + 7.38iT - 43T^{2} \) |
| 47 | \( 1 + 5.59iT - 47T^{2} \) |
| 53 | \( 1 + 12.3iT - 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 14.7T + 61T^{2} \) |
| 67 | \( 1 - 0.719iT - 67T^{2} \) |
| 71 | \( 1 + 5.38T + 71T^{2} \) |
| 73 | \( 1 - 10.6iT - 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 1.99iT - 83T^{2} \) |
| 89 | \( 1 - 6.30T + 89T^{2} \) |
| 97 | \( 1 - 0.585iT - 97T^{2} \) |
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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
−8.318673147407683211545021508643, −7.13814813142277780892837426662, −6.61432915190049310837853480490, −5.80056611316204513114925570197, −4.85639513724801750058305707054, −4.17639907402897738481425796353, −3.63785323451695392135320298194, −2.08032192370504944087927507787, −1.38870943490184239679671564790, −0.097624501961404916859034501246,
1.80349727219319651280828942489, 2.64601321855580880524870390533, 3.09050642332758118899066099427, 4.37096371670596406160316280411, 5.41565789041014954775675120377, 5.80218559541710461311456422091, 6.54932697762765868080412474650, 7.39415812830456701721011575145, 8.058294746021520123033310834725, 9.102199370772825389115553253257
