L(s) = 1 | + 3-s − 5-s + 7-s + 9-s − 11-s + 13-s − 15-s + 17-s + 21-s − 23-s + 25-s + 27-s − 33-s − 35-s − 37-s + 39-s + 41-s − 45-s + 51-s + 53-s + 55-s + 2·59-s − 61-s + 63-s − 65-s − 2·67-s − 69-s + ⋯ |
L(s) = 1 | + 3-s − 5-s + 7-s + 9-s − 11-s + 13-s − 15-s + 17-s + 21-s − 23-s + 25-s + 27-s − 33-s − 35-s − 37-s + 39-s + 41-s − 45-s + 51-s + 53-s + 55-s + 2·59-s − 61-s + 63-s − 65-s − 2·67-s − 69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.729990078\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.729990078\) |
\(L(1)\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( ( 1 - T )^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 + T )^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−8.570737321717064129093260951153, −8.069921031985728068978234735272, −7.73753885458730224796276823050, −6.93878552104349677370007872837, −5.72672513555312886640567640736, −4.86007127485413751093750792151, −4.02114592491496263531006856858, −3.39446060667188153191247913764, −2.38276126753312825716323569126, −1.25206130511958576504027275991,
1.25206130511958576504027275991, 2.38276126753312825716323569126, 3.39446060667188153191247913764, 4.02114592491496263531006856858, 4.86007127485413751093750792151, 5.72672513555312886640567640736, 6.93878552104349677370007872837, 7.73753885458730224796276823050, 8.069921031985728068978234735272, 8.570737321717064129093260951153