L(s) = 1 | − 1.41·3-s − 5-s − 2.82·7-s − 0.999·9-s − 1.41·11-s + 13-s + 1.41·15-s − 6·17-s + 7.07·19-s + 4.00·21-s + 7.07·23-s + 25-s + 5.65·27-s + 8·29-s + 4.24·31-s + 2.00·33-s + 2.82·35-s − 1.41·39-s − 6·41-s + 4.24·43-s + 0.999·45-s − 2.82·47-s + 1.00·49-s + 8.48·51-s + 2·53-s + 1.41·55-s − 10.0·57-s + ⋯ |
L(s) = 1 | − 0.816·3-s − 0.447·5-s − 1.06·7-s − 0.333·9-s − 0.426·11-s + 0.277·13-s + 0.365·15-s − 1.45·17-s + 1.62·19-s + 0.872·21-s + 1.47·23-s + 0.200·25-s + 1.08·27-s + 1.48·29-s + 0.762·31-s + 0.348·33-s + 0.478·35-s − 0.226·39-s − 0.937·41-s + 0.646·43-s + 0.149·45-s − 0.412·47-s + 0.142·49-s + 1.18·51-s + 0.274·53-s + 0.190·55-s − 1.32·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 23 | \( 1 - 7.07T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4.24T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 16T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.077197676206656156478148355301, −7.01881999629622241026053689016, −6.65573217645546071800689555319, −5.87817218314399290292507425768, −5.06710292980266947720374365581, −4.43971365419013320651594151726, −3.16633172331456865124510715806, −2.80905182220290743280536444944, −1.04295956697045987344376313603, 0,
1.04295956697045987344376313603, 2.80905182220290743280536444944, 3.16633172331456865124510715806, 4.43971365419013320651594151726, 5.06710292980266947720374365581, 5.87817218314399290292507425768, 6.65573217645546071800689555319, 7.01881999629622241026053689016, 8.077197676206656156478148355301