| L(s) = 1 | − 13.3·5-s + 14.3·7-s + 39.0·11-s − 76.7·13-s + 62.4·17-s + 39.7·19-s − 128.·23-s + 53.4·25-s − 64.9·29-s + 9.13·31-s − 191.·35-s + 319.·37-s − 17.5·41-s + 450.·43-s − 581.·47-s − 136.·49-s − 329.·53-s − 521.·55-s − 241.·59-s + 497.·61-s + 1.02e3·65-s + 578.·67-s − 660.·71-s + 696.·73-s + 561.·77-s − 730.·79-s + 1.09e3·83-s + ⋯ |
| L(s) = 1 | − 1.19·5-s + 0.775·7-s + 1.07·11-s − 1.63·13-s + 0.890·17-s + 0.480·19-s − 1.16·23-s + 0.427·25-s − 0.415·29-s + 0.0529·31-s − 0.926·35-s + 1.41·37-s − 0.0669·41-s + 1.59·43-s − 1.80·47-s − 0.398·49-s − 0.853·53-s − 1.27·55-s − 0.533·59-s + 1.04·61-s + 1.95·65-s + 1.05·67-s − 1.10·71-s + 1.11·73-s + 0.830·77-s − 1.04·79-s + 1.44·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.575485235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.575485235\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 13.3T + 125T^{2} \) |
| 7 | \( 1 - 14.3T + 343T^{2} \) |
| 11 | \( 1 - 39.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 76.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 62.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 39.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 128.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 64.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 9.13T + 2.97e4T^{2} \) |
| 37 | \( 1 - 319.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 17.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 450.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 581.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 329.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 241.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 497.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 578.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 660.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 696.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 730.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.09e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 317.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.48e3T + 9.12e5T^{2} \) |
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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
−9.422804602741926958495143010260, −8.145296130985831226225685035136, −7.78831046861489376111385267835, −7.07122193975153225434139187955, −5.92975594918966908579538283903, −4.83806714372025385014111813699, −4.19214927457556576083441615566, −3.24920259618646333813096102988, −1.92971455292356922333819739672, −0.63441062142877069893173997154,
0.63441062142877069893173997154, 1.92971455292356922333819739672, 3.24920259618646333813096102988, 4.19214927457556576083441615566, 4.83806714372025385014111813699, 5.92975594918966908579538283903, 7.07122193975153225434139187955, 7.78831046861489376111385267835, 8.145296130985831226225685035136, 9.422804602741926958495143010260
