| L(s) = 1 | + 1.18·3-s − 18.9·5-s − 9.32·7-s − 25.5·9-s + 39.8·11-s − 22.4·15-s − 92.6·17-s − 128.·19-s − 11.0·21-s − 158.·23-s + 235.·25-s − 62.3·27-s − 126.·29-s + 189.·31-s + 47.1·33-s + 177.·35-s + 53.7·37-s + 136.·41-s − 518.·43-s + 485.·45-s − 309.·47-s − 256.·49-s − 109.·51-s − 149.·53-s − 755.·55-s − 152.·57-s − 74.3·59-s + ⋯ |
| L(s) = 1 | + 0.227·3-s − 1.69·5-s − 0.503·7-s − 0.948·9-s + 1.09·11-s − 0.387·15-s − 1.32·17-s − 1.55·19-s − 0.114·21-s − 1.44·23-s + 1.88·25-s − 0.444·27-s − 0.810·29-s + 1.09·31-s + 0.248·33-s + 0.854·35-s + 0.238·37-s + 0.518·41-s − 1.83·43-s + 1.60·45-s − 0.961·47-s − 0.746·49-s − 0.301·51-s − 0.387·53-s − 1.85·55-s − 0.353·57-s − 0.164·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3836216696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3836216696\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.18T + 27T^{2} \) |
| 5 | \( 1 + 18.9T + 125T^{2} \) |
| 7 | \( 1 + 9.32T + 343T^{2} \) |
| 11 | \( 1 - 39.8T + 1.33e3T^{2} \) |
| 17 | \( 1 + 92.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 128.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 158.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 126.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 189.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 53.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 136.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 518.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 309.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 149.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 74.3T + 2.05e5T^{2} \) |
| 61 | \( 1 - 99.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 435.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 827.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 981.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 299.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 169.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 265.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 88.9T + 9.12e5T^{2} \) |
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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
−8.957731846739305017059019796244, −8.415978212537576662602916854825, −7.83486608790089527628351696854, −6.63472267819900511832372571556, −6.28491183337120296248273517455, −4.69275062316536028500538370536, −3.99031262054212261143474042460, −3.32175210367181366280259455666, −2.08534343408821721454341294932, −0.28802171303957937644996188159,
0.28802171303957937644996188159, 2.08534343408821721454341294932, 3.32175210367181366280259455666, 3.99031262054212261143474042460, 4.69275062316536028500538370536, 6.28491183337120296248273517455, 6.63472267819900511832372571556, 7.83486608790089527628351696854, 8.415978212537576662602916854825, 8.957731846739305017059019796244
