| L(s) = 1 | + i·2-s − 4-s − i·7-s − i·8-s + 4·11-s − 6i·13-s + 14-s + 16-s − 4i·17-s − 6·19-s + 4i·22-s + 6·26-s + i·28-s − 6·29-s − 4·31-s + i·32-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.377i·7-s − 0.353i·8-s + 1.20·11-s − 1.66i·13-s + 0.267·14-s + 0.250·16-s − 0.970i·17-s − 1.37·19-s + 0.852i·22-s + 1.17·26-s + 0.188i·28-s − 1.11·29-s − 0.718·31-s + 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5982553931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5982553931\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + 10iT - 47T^{2} \) |
| 53 | \( 1 - 14iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 6iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.452638401522782125996941855119, −7.57417570802799638399813567576, −6.99427010476788893980956406078, −6.21935800601662952174837603363, −5.49338183265968951560534288483, −4.64343749736367863339587409196, −3.80749737426478249365233225326, −2.95592546566307436550590342399, −1.47430329652078000632173265111, −0.18014242094875388600744436718,
1.69788553547637432017326320021, 2.02369591564810194406885870016, 3.55794752597154190458651212811, 4.04741547347709638169950482970, 4.84336855542512028900222161397, 6.03471319516934027341951167081, 6.51695129391570008242047856145, 7.43204772394178331423441551924, 8.551688120957419184863446133736, 8.989944523851944745677400490040
