L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 3·11-s + 13-s + 15-s + 17-s + 19-s − 21-s − 5·23-s − 4·25-s − 27-s − 6·29-s − 4·31-s − 3·33-s − 35-s − 10·37-s − 39-s + 5·41-s + 9·43-s − 45-s − 4·47-s + 49-s − 51-s − 3·55-s − 57-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 0.258·15-s + 0.242·17-s + 0.229·19-s − 0.218·21-s − 1.04·23-s − 4/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.522·33-s − 0.169·35-s − 1.64·37-s − 0.160·39-s + 0.780·41-s + 1.37·43-s − 0.149·45-s − 0.583·47-s + 1/7·49-s − 0.140·51-s − 0.404·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.60345668485865132301907341599, −7.19935695690006767719251320438, −6.19333314538255023241147208764, −5.72805232448678828392387423741, −4.87735940774901567569307614632, −3.95006099034600774434118180846, −3.60627691480015650617878795170, −2.14413307946240294844137451565, −1.29260892932866668162717223337, 0,
1.29260892932866668162717223337, 2.14413307946240294844137451565, 3.60627691480015650617878795170, 3.95006099034600774434118180846, 4.87735940774901567569307614632, 5.72805232448678828392387423741, 6.19333314538255023241147208764, 7.19935695690006767719251320438, 7.60345668485865132301907341599