L(s) = 1 | − 3-s − 5-s + 9-s + 3·11-s + 13-s + 15-s − 7·19-s − 5·23-s + 25-s − 27-s + 6·31-s − 3·33-s + 3·37-s − 39-s − 3·41-s + 8·43-s − 45-s − 47-s + 5·53-s − 3·55-s + 7·57-s − 4·59-s − 8·61-s − 65-s + 5·69-s − 6·71-s − 14·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 0.258·15-s − 1.60·19-s − 1.04·23-s + 1/5·25-s − 0.192·27-s + 1.07·31-s − 0.522·33-s + 0.493·37-s − 0.160·39-s − 0.468·41-s + 1.21·43-s − 0.149·45-s − 0.145·47-s + 0.686·53-s − 0.404·55-s + 0.927·57-s − 0.520·59-s − 1.02·61-s − 0.124·65-s + 0.601·69-s − 0.712·71-s − 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 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
−7.67042796079944065350827155902, −6.99488418754050730925377694443, −6.11320828871754511922270552449, −5.95175758332091253185486986935, −4.47529916529308320676973238715, −4.37362310519535493772865036625, −3.37450905718996629601741580375, −2.24150651846838351084631203694, −1.21573778774701355548847083559, 0,
1.21573778774701355548847083559, 2.24150651846838351084631203694, 3.37450905718996629601741580375, 4.37362310519535493772865036625, 4.47529916529308320676973238715, 5.95175758332091253185486986935, 6.11320828871754511922270552449, 6.99488418754050730925377694443, 7.67042796079944065350827155902