| L(s) = 1 | + 8.74·3-s − 21.7·5-s − 20.7·7-s + 49.5·9-s − 28.6·11-s + 13·13-s − 189.·15-s − 111.·17-s − 17.6·19-s − 181.·21-s − 89.6·23-s + 346.·25-s + 197.·27-s + 27.0·29-s − 61.4·31-s − 250.·33-s + 450.·35-s + 55.5·37-s + 113.·39-s + 281.·41-s − 276.·43-s − 1.07e3·45-s + 89.4·47-s + 87.5·49-s − 972.·51-s − 127.·53-s + 622.·55-s + ⋯ |
| L(s) = 1 | + 1.68·3-s − 1.94·5-s − 1.12·7-s + 1.83·9-s − 0.785·11-s + 0.277·13-s − 3.26·15-s − 1.58·17-s − 0.213·19-s − 1.88·21-s − 0.812·23-s + 2.76·25-s + 1.40·27-s + 0.172·29-s − 0.356·31-s − 1.32·33-s + 2.17·35-s + 0.247·37-s + 0.466·39-s + 1.07·41-s − 0.980·43-s − 3.56·45-s + 0.277·47-s + 0.255·49-s − 2.67·51-s − 0.330·53-s + 1.52·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 3 | \( 1 - 8.74T + 27T^{2} \) |
| 5 | \( 1 + 21.7T + 125T^{2} \) |
| 7 | \( 1 + 20.7T + 343T^{2} \) |
| 11 | \( 1 + 28.6T + 1.33e3T^{2} \) |
| 17 | \( 1 + 111.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 17.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 89.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 27.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 61.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 55.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 281.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 276.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 89.4T + 1.03e5T^{2} \) |
| 53 | \( 1 + 127.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 132.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 674.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 398.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 767.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 533.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 579.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.07e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 586.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 6.09T + 9.12e5T^{2} \) |
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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
−11.49253911377865686712678052079, −10.37141612227359368725096664094, −9.133453310859181653495938711646, −8.390175285537483189256238891079, −7.65411637668296042350842764519, −6.72954970611258982521419053735, −4.36568419750327829356817823139, −3.58528592580867839771436769303, −2.61448972412324647225335342863, 0,
2.61448972412324647225335342863, 3.58528592580867839771436769303, 4.36568419750327829356817823139, 6.72954970611258982521419053735, 7.65411637668296042350842764519, 8.390175285537483189256238891079, 9.133453310859181653495938711646, 10.37141612227359368725096664094, 11.49253911377865686712678052079
