| L(s) = 1 | − 1.75·3-s − 0.457·5-s + 7-s + 0.0972·9-s − 5.74·11-s + 0.804·15-s − 2.95·17-s + 5.73·19-s − 1.75·21-s − 3.76·23-s − 4.79·25-s + 5.10·27-s − 3.90·29-s + 5.27·31-s + 10.1·33-s − 0.457·35-s − 4.96·37-s − 4.25·41-s − 6.07·43-s − 0.0444·45-s + 0.377·47-s + 49-s + 5.20·51-s − 1.05·53-s + 2.62·55-s − 10.0·57-s − 12.0·59-s + ⋯ |
| L(s) = 1 | − 1.01·3-s − 0.204·5-s + 0.377·7-s + 0.0324·9-s − 1.73·11-s + 0.207·15-s − 0.717·17-s + 1.31·19-s − 0.384·21-s − 0.784·23-s − 0.958·25-s + 0.983·27-s − 0.726·29-s + 0.948·31-s + 1.76·33-s − 0.0772·35-s − 0.816·37-s − 0.664·41-s − 0.926·43-s − 0.00662·45-s + 0.0551·47-s + 0.142·49-s + 0.728·51-s − 0.144·53-s + 0.354·55-s − 1.33·57-s − 1.57·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4465280021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4465280021\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 1.75T + 3T^{2} \) |
| 5 | \( 1 + 0.457T + 5T^{2} \) |
| 11 | \( 1 + 5.74T + 11T^{2} \) |
| 17 | \( 1 + 2.95T + 17T^{2} \) |
| 19 | \( 1 - 5.73T + 19T^{2} \) |
| 23 | \( 1 + 3.76T + 23T^{2} \) |
| 29 | \( 1 + 3.90T + 29T^{2} \) |
| 31 | \( 1 - 5.27T + 31T^{2} \) |
| 37 | \( 1 + 4.96T + 37T^{2} \) |
| 41 | \( 1 + 4.25T + 41T^{2} \) |
| 43 | \( 1 + 6.07T + 43T^{2} \) |
| 47 | \( 1 - 0.377T + 47T^{2} \) |
| 53 | \( 1 + 1.05T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 0.425T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 9.05T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 9.93T + 83T^{2} \) |
| 89 | \( 1 + 9.60T + 89T^{2} \) |
| 97 | \( 1 - 9.16T + 97T^{2} \) |
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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
−7.66815331133727436841011802289, −7.05930090612375951268544687505, −6.15443283046659702627661906080, −5.57102672165453040920071478869, −5.08474742768575085759905430023, −4.47276226170703398954049014216, −3.40924842289631939292170064122, −2.61139050893100833593331206967, −1.64968037221563561315702863209, −0.32874051185392578013630460234,
0.32874051185392578013630460234, 1.64968037221563561315702863209, 2.61139050893100833593331206967, 3.40924842289631939292170064122, 4.47276226170703398954049014216, 5.08474742768575085759905430023, 5.57102672165453040920071478869, 6.15443283046659702627661906080, 7.05930090612375951268544687505, 7.66815331133727436841011802289
