L(s) = 1 | − 2.72·2-s + 3-s + 5.44·4-s − 5-s − 2.72·6-s + 4.68·7-s − 9.40·8-s + 9-s + 2.72·10-s + 2.58·11-s + 5.44·12-s + 4.72·13-s − 12.7·14-s − 15-s + 14.7·16-s − 1.85·17-s − 2.72·18-s − 2.34·19-s − 5.44·20-s + 4.68·21-s − 7.04·22-s + 5.03·23-s − 9.40·24-s + 25-s − 12.8·26-s + 27-s + 25.5·28-s + ⋯ |
L(s) = 1 | − 1.92·2-s + 0.577·3-s + 2.72·4-s − 0.447·5-s − 1.11·6-s + 1.77·7-s − 3.32·8-s + 0.333·9-s + 0.862·10-s + 0.778·11-s + 1.57·12-s + 1.31·13-s − 3.41·14-s − 0.258·15-s + 3.69·16-s − 0.450·17-s − 0.643·18-s − 0.538·19-s − 1.21·20-s + 1.02·21-s − 1.50·22-s + 1.04·23-s − 1.91·24-s + 0.200·25-s − 2.52·26-s + 0.192·27-s + 4.82·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.424173641\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.424173641\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 7 | \( 1 - 4.68T + 7T^{2} \) |
| 11 | \( 1 - 2.58T + 11T^{2} \) |
| 13 | \( 1 - 4.72T + 13T^{2} \) |
| 17 | \( 1 + 1.85T + 17T^{2} \) |
| 19 | \( 1 + 2.34T + 19T^{2} \) |
| 23 | \( 1 - 5.03T + 23T^{2} \) |
| 29 | \( 1 + 3.14T + 29T^{2} \) |
| 31 | \( 1 + 2.55T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 + 2.32T + 43T^{2} \) |
| 47 | \( 1 + 6.42T + 47T^{2} \) |
| 53 | \( 1 - 5.44T + 53T^{2} \) |
| 59 | \( 1 + 1.63T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 - 2.51T + 67T^{2} \) |
| 71 | \( 1 + 6.26T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 6.58T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 - 7.85T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−8.349701518407730544564229896021, −7.61137577047963000102006788529, −7.10300438039717670187744822882, −6.37033835433983673848448112064, −5.39139132098564440510725785036, −4.23775889816111158083160835940, −3.42081576335670033121913213369, −2.24798956087015294480960326911, −1.59300053961489334452497929529, −0.892264595876107693034959975438,
0.892264595876107693034959975438, 1.59300053961489334452497929529, 2.24798956087015294480960326911, 3.42081576335670033121913213369, 4.23775889816111158083160835940, 5.39139132098564440510725785036, 6.37033835433983673848448112064, 7.10300438039717670187744822882, 7.61137577047963000102006788529, 8.349701518407730544564229896021