L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 11-s + 2·13-s + 15-s − 7·17-s + 3·19-s − 21-s + 3·23-s + 25-s − 27-s − 5·29-s + 33-s − 35-s + 2·37-s − 2·39-s + 43-s − 45-s + 8·47-s + 49-s + 7·51-s − 9·53-s + 55-s − 3·57-s − 9·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.258·15-s − 1.69·17-s + 0.688·19-s − 0.218·21-s + 0.625·23-s + 1/5·25-s − 0.192·27-s − 0.928·29-s + 0.174·33-s − 0.169·35-s + 0.328·37-s − 0.320·39-s + 0.152·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.980·51-s − 1.23·53-s + 0.134·55-s − 0.397·57-s − 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−7.78225500677182705297332048355, −7.30336231191537439889256836716, −6.46883887331436998124488785674, −5.78529533181259764992373514999, −4.90610131514061825327369965470, −4.34764133494520801888026569575, −3.44425184165626550095176281670, −2.37117777658916015617857375561, −1.26393328193811222436942412170, 0,
1.26393328193811222436942412170, 2.37117777658916015617857375561, 3.44425184165626550095176281670, 4.34764133494520801888026569575, 4.90610131514061825327369965470, 5.78529533181259764992373514999, 6.46883887331436998124488785674, 7.30336231191537439889256836716, 7.78225500677182705297332048355