L(s) = 1 | + 3-s + 5-s + 3.20·7-s + 9-s + 5.20·11-s + 5.45·13-s + 15-s + 4·17-s + 19-s + 3.20·21-s − 8.65·23-s + 25-s + 27-s − 3.20·29-s − 2·31-s + 5.20·33-s + 3.20·35-s − 11.8·37-s + 5.45·39-s − 9.85·41-s − 3.45·43-s + 45-s + 8.65·47-s + 3.25·49-s + 4·51-s − 10.6·53-s + 5.20·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.21·7-s + 0.333·9-s + 1.56·11-s + 1.51·13-s + 0.258·15-s + 0.970·17-s + 0.229·19-s + 0.698·21-s − 1.80·23-s + 0.200·25-s + 0.192·27-s − 0.594·29-s − 0.359·31-s + 0.905·33-s + 0.541·35-s − 1.94·37-s + 0.873·39-s − 1.53·41-s − 0.526·43-s + 0.149·45-s + 1.26·47-s + 0.464·49-s + 0.560·51-s − 1.46·53-s + 0.701·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.275187771\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.275187771\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 3.20T + 7T^{2} \) |
| 11 | \( 1 - 5.20T + 11T^{2} \) |
| 13 | \( 1 - 5.45T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 23 | \( 1 + 8.65T + 23T^{2} \) |
| 29 | \( 1 + 3.20T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 11.8T + 37T^{2} \) |
| 41 | \( 1 + 9.85T + 41T^{2} \) |
| 43 | \( 1 + 3.45T + 43T^{2} \) |
| 47 | \( 1 - 8.65T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 - 3.74T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 6.90T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 6.15T + 79T^{2} \) |
| 83 | \( 1 + 8.40T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 0.142T + 97T^{2} \) |
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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
−8.873075671607532913787849384293, −8.360881948826007078560473697525, −7.65785342906162182691483641053, −6.64845931668855687924510046732, −5.92843587225867296271803274547, −5.04697021534655749578656915407, −3.91463183154030717323806723555, −3.47446918324216801461481856338, −1.77513826055931309509211154237, −1.45135511230001580958708003342,
1.45135511230001580958708003342, 1.77513826055931309509211154237, 3.47446918324216801461481856338, 3.91463183154030717323806723555, 5.04697021534655749578656915407, 5.92843587225867296271803274547, 6.64845931668855687924510046732, 7.65785342906162182691483641053, 8.360881948826007078560473697525, 8.873075671607532913787849384293