| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s − 3·11-s − 12-s − 6·13-s − 14-s + 15-s + 16-s − 17-s + 18-s + 3·19-s − 20-s + 21-s − 3·22-s + 23-s − 24-s + 25-s − 6·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.904·11-s − 0.288·12-s − 1.66·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.688·19-s − 0.223·20-s + 0.218·21-s − 0.639·22-s + 0.208·23-s − 0.204·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.250106282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.250106282\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.91867174053478, −12.69721951482539, −12.44267657880321, −11.67915382747836, −11.48253077223422, −10.98972899336623, −10.31881683022254, −10.03127639014945, −9.503442146320926, −9.012906524379103, −8.076634061163720, −7.812912977461408, −7.279365970324017, −6.864432687593563, −6.391643676583138, −5.557774175438424, −5.361837592186921, −4.884432652903239, −4.182807957748514, −3.925188233232820, −2.967963912396246, −2.577334902562545, −2.165085752606153, −0.9886157251459698, −0.4590487679652049,
0.4590487679652049, 0.9886157251459698, 2.165085752606153, 2.577334902562545, 2.967963912396246, 3.925188233232820, 4.182807957748514, 4.884432652903239, 5.361837592186921, 5.557774175438424, 6.391643676583138, 6.864432687593563, 7.279365970324017, 7.812912977461408, 8.076634061163720, 9.012906524379103, 9.503442146320926, 10.03127639014945, 10.31881683022254, 10.98972899336623, 11.48253077223422, 11.67915382747836, 12.44267657880321, 12.69721951482539, 12.91867174053478
