| L(s) = 1 | − 2-s − 3-s + 4-s + 3·5-s + 6-s − 8-s + 9-s − 3·10-s − 11-s − 12-s − 2·13-s − 3·15-s + 16-s − 3·17-s − 18-s − 2·19-s + 3·20-s + 22-s + 3·23-s + 24-s + 4·25-s + 2·26-s − 27-s − 6·29-s + 3·30-s + 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.301·11-s − 0.288·12-s − 0.554·13-s − 0.774·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.458·19-s + 0.670·20-s + 0.213·22-s + 0.625·23-s + 0.204·24-s + 4/5·25-s + 0.392·26-s − 0.192·27-s − 1.11·29-s + 0.547·30-s + 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.257285605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.257285605\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−8.832180172697162948145836741478, −7.943073598331531796074729901310, −7.02459876658352966555909048207, −6.49626868640388813351839269616, −5.67199925636180801239165837803, −5.13599190165420974719855633781, −4.02964545591551333118333607535, −2.56485651029984601868018839076, −2.02572365669600761750685686373, −0.76226570382701843132880365607,
0.76226570382701843132880365607, 2.02572365669600761750685686373, 2.56485651029984601868018839076, 4.02964545591551333118333607535, 5.13599190165420974719855633781, 5.67199925636180801239165837803, 6.49626868640388813351839269616, 7.02459876658352966555909048207, 7.943073598331531796074729901310, 8.832180172697162948145836741478
