| L(s) = 1 | − 0.732i·7-s + 1.73·11-s − 1.46i·13-s − 1.26i·17-s − 2.46·19-s + 3.46i·23-s − 4.26·29-s − 7.92·31-s − 4.19i·37-s + 0.803·41-s + 6.73i·43-s − 4.73i·47-s + 6.46·49-s + 10.7i·53-s − 4.26·59-s + ⋯ |
| L(s) = 1 | − 0.276i·7-s + 0.522·11-s − 0.406i·13-s − 0.307i·17-s − 0.565·19-s + 0.722i·23-s − 0.792·29-s − 1.42·31-s − 0.689i·37-s + 0.125·41-s + 1.02i·43-s − 0.690i·47-s + 0.923·49-s + 1.47i·53-s − 0.555·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8043169040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8043169040\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 0.732iT - 7T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 + 1.46iT - 13T^{2} \) |
| 17 | \( 1 + 1.26iT - 17T^{2} \) |
| 19 | \( 1 + 2.46T + 19T^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 + 4.26T + 29T^{2} \) |
| 31 | \( 1 + 7.92T + 31T^{2} \) |
| 37 | \( 1 + 4.19iT - 37T^{2} \) |
| 41 | \( 1 - 0.803T + 41T^{2} \) |
| 43 | \( 1 - 6.73iT - 43T^{2} \) |
| 47 | \( 1 + 4.73iT - 47T^{2} \) |
| 53 | \( 1 - 10.7iT - 53T^{2} \) |
| 59 | \( 1 + 4.26T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 14.3iT - 67T^{2} \) |
| 71 | \( 1 - 0.803T + 71T^{2} \) |
| 73 | \( 1 - 10.1iT - 73T^{2} \) |
| 79 | \( 1 + 6.39T + 79T^{2} \) |
| 83 | \( 1 + 9.12iT - 83T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 - 2.73iT - 97T^{2} \) |
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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
−7.86419007211873361848351230756, −7.40601667554485144387382239761, −6.76843687147166538265192234201, −5.84684302046422547312797541915, −5.43373247788282536580030267928, −4.38056870104588391142390359373, −3.83965080749752124721595933566, −2.99951132303552983284904893186, −2.02333429146801692597178282020, −1.08561613217301888949976881122,
0.19290917447958255108560666359, 1.59060936046664083177389947721, 2.25194070438566794164048200024, 3.34179194150733243741301246630, 4.00897604053955738836722710170, 4.76016505369932647820947080055, 5.57102585267548162113678575558, 6.26707533240463514297015927530, 6.85319282589035744324347460079, 7.58532059220180528423111794867
