| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s + 2·11-s + 12-s + 6·13-s − 14-s + 16-s + 4·17-s + 18-s − 6·19-s − 21-s + 2·22-s − 8·23-s + 24-s + 6·26-s + 27-s − 28-s + 6·29-s − 2·31-s + 32-s + 2·33-s + 4·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.603·11-s + 0.288·12-s + 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 1.37·19-s − 0.218·21-s + 0.426·22-s − 1.66·23-s + 0.204·24-s + 1.17·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.359·31-s + 0.176·32-s + 0.348·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.184789179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.184789179\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−10.04443335843472068951038682726, −8.936983560301573627344332758630, −8.312948613577623088028079646706, −7.37232550640409553455815986114, −6.21997249752887011936324535023, −5.92502214493828437001251745493, −4.28320630078127508245235828197, −3.81978824771939003300116575643, −2.72783326557746750240871274336, −1.42704273090331691804140546291,
1.42704273090331691804140546291, 2.72783326557746750240871274336, 3.81978824771939003300116575643, 4.28320630078127508245235828197, 5.92502214493828437001251745493, 6.21997249752887011936324535023, 7.37232550640409553455815986114, 8.312948613577623088028079646706, 8.936983560301573627344332758630, 10.04443335843472068951038682726
