| L(s) = 1 | + 0.414·2-s − 1.82·4-s − 3.41·5-s − 0.585·7-s − 1.58·8-s − 1.41·10-s − 2.82·11-s − 0.242·14-s + 3·16-s − 4.58·17-s + 2.24·19-s + 6.24·20-s − 1.17·22-s − 23-s + 6.65·25-s + 1.07·28-s + 8.48·29-s − 8.48·31-s + 4.41·32-s − 1.89·34-s + 2·35-s + 0.828·37-s + 0.928·38-s + 5.41·40-s − 9.65·41-s − 10.2·43-s + 5.17·44-s + ⋯ |
| L(s) = 1 | + 0.292·2-s − 0.914·4-s − 1.52·5-s − 0.221·7-s − 0.560·8-s − 0.447·10-s − 0.852·11-s − 0.0648·14-s + 0.750·16-s − 1.11·17-s + 0.514·19-s + 1.39·20-s − 0.249·22-s − 0.208·23-s + 1.33·25-s + 0.202·28-s + 1.57·29-s − 1.52·31-s + 0.780·32-s − 0.325·34-s + 0.338·35-s + 0.136·37-s + 0.150·38-s + 0.856·40-s − 1.50·41-s − 1.56·43-s + 0.779·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 5 | \( 1 + 3.41T + 5T^{2} \) |
| 7 | \( 1 + 0.585T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 4.58T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 - 0.828T + 37T^{2} \) |
| 41 | \( 1 + 9.65T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 9.07T + 53T^{2} \) |
| 59 | \( 1 - 3.65T + 59T^{2} \) |
| 61 | \( 1 - 4.82T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 2.34T + 71T^{2} \) |
| 73 | \( 1 + 9.31T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 + 1.17T + 83T^{2} \) |
| 89 | \( 1 - 1.07T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−12.04306302795660426804772859266, −11.09915204036058115924477961631, −9.964339978482274388033449522212, −8.698348566219021670728500586163, −8.043276254295977733764515035215, −6.87087535512029452118997036170, −5.23283621324211346579253115359, −4.27328667449444397289866024716, −3.21105387653454934310381798050, 0,
3.21105387653454934310381798050, 4.27328667449444397289866024716, 5.23283621324211346579253115359, 6.87087535512029452118997036170, 8.043276254295977733764515035215, 8.698348566219021670728500586163, 9.964339978482274388033449522212, 11.09915204036058115924477961631, 12.04306302795660426804772859266
