L(s) = 1 | + 3-s + 9-s − 11-s − 13-s − 6·17-s − 3·19-s − 9·23-s + 27-s − 10·31-s − 33-s + 3·37-s − 39-s + 9·41-s + 10·43-s + 9·47-s − 6·51-s − 9·53-s − 3·57-s − 12·59-s − 6·61-s − 2·67-s − 9·69-s + 6·71-s − 4·73-s + 10·79-s + 81-s − 6·83-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 1.45·17-s − 0.688·19-s − 1.87·23-s + 0.192·27-s − 1.79·31-s − 0.174·33-s + 0.493·37-s − 0.160·39-s + 1.40·41-s + 1.52·43-s + 1.31·47-s − 0.840·51-s − 1.23·53-s − 0.397·57-s − 1.56·59-s − 0.768·61-s − 0.244·67-s − 1.08·69-s + 0.712·71-s − 0.468·73-s + 1.12·79-s + 1/9·81-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.516552832\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.516552832\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−15.13754855078766, −14.60645728084408, −14.05351950351648, −13.73242400950576, −12.96471641346598, −12.59612346733834, −12.16355596946197, −11.25873291784676, −10.80950869158281, −10.46471631302860, −9.476994459031056, −9.294997013984009, −8.688686005242559, −7.945793448153380, −7.589311554163267, −6.988551788559489, −6.055472868040156, −5.923823059746470, −4.817558777807441, −4.249814628642066, −3.851882158389976, −2.864684084270870, −2.220543104168214, −1.776833342517290, −0.4288159517232632,
0.4288159517232632, 1.776833342517290, 2.220543104168214, 2.864684084270870, 3.851882158389976, 4.249814628642066, 4.817558777807441, 5.923823059746470, 6.055472868040156, 6.988551788559489, 7.589311554163267, 7.945793448153380, 8.688686005242559, 9.294997013984009, 9.476994459031056, 10.46471631302860, 10.80950869158281, 11.25873291784676, 12.16355596946197, 12.59612346733834, 12.96471641346598, 13.73242400950576, 14.05351950351648, 14.60645728084408, 15.13754855078766