| L(s) = 1 | + 2-s + 4-s − 5i·7-s + 8-s + 3·9-s + 3·11-s − 2·13-s − 5i·14-s + 16-s − 17-s + 3·18-s − 2i·19-s + 3·22-s − 6·23-s − 2·26-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.88i·7-s + 0.353·8-s + 9-s + 0.904·11-s − 0.554·13-s − 1.33i·14-s + 0.250·16-s − 0.242·17-s + 0.707·18-s − 0.458i·19-s + 0.639·22-s − 1.25·23-s − 0.392·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.294 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.294 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.859716940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.859716940\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + (6 + i)T \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 + 5iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 5iT - 29T^{2} \) |
| 31 | \( 1 - iT - 31T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 - 11T + 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 + 7iT - 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 4iT - 89T^{2} \) |
| 97 | \( 1 + 11T + 97T^{2} \) |
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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
−9.309599800347654860591978449518, −7.947640534138135920507473756675, −7.30877147646708006609159942951, −6.81318215614629652702502073436, −5.96676924548532722673408752191, −4.53867238438763079385203332593, −4.27798398581720294187128543463, −3.49614269472130565669986720457, −2.00602650260678251321626644522, −0.851006944572910665417832137267,
1.69417294331642415473009116422, 2.45054938670335094397977526086, 3.62683743529594789690438247827, 4.50061658424296055028328432439, 5.41401325149031698123111522902, 6.07253757774285045766794578071, 6.83921422411362489763174019776, 7.75049846147459723594163771463, 8.713008769756002695167083309848, 9.365433901307916757036876238777
