| L(s) = 1 | − 2-s + 3-s + 4-s − 1.41·5-s − 6-s − 8-s + 9-s + 1.41·10-s + 2·11-s + 12-s − 1.41·15-s + 16-s − 4·17-s − 18-s − 2.58·19-s − 1.41·20-s − 2·22-s + 23-s − 24-s − 2.99·25-s + 27-s + 7.65·29-s + 1.41·30-s − 8.24·31-s − 32-s + 2·33-s + 4·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.632·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.447·10-s + 0.603·11-s + 0.288·12-s − 0.365·15-s + 0.250·16-s − 0.970·17-s − 0.235·18-s − 0.593·19-s − 0.316·20-s − 0.426·22-s + 0.208·23-s − 0.204·24-s − 0.599·25-s + 0.192·27-s + 1.42·29-s + 0.258·30-s − 1.48·31-s − 0.176·32-s + 0.348·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 2.58T + 19T^{2} \) |
| 29 | \( 1 - 7.65T + 29T^{2} \) |
| 31 | \( 1 + 8.24T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 - 3.07T + 47T^{2} \) |
| 53 | \( 1 + 9.31T + 53T^{2} \) |
| 59 | \( 1 - 1.65T + 59T^{2} \) |
| 61 | \( 1 - 4.24T + 61T^{2} \) |
| 67 | \( 1 - 1.65T + 67T^{2} \) |
| 71 | \( 1 + 4.34T + 71T^{2} \) |
| 73 | \( 1 + 9.89T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 9.17T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−7.64005798105140620305917364890, −7.16804590368143171155605018595, −6.44474571909013809095829520191, −5.68881968275444669872880949719, −4.42100420657712233064020254978, −4.04967922504011595429373759486, −3.02113625063977938342676736027, −2.24895683337272173927983718538, −1.26673939202022297364452388972, 0,
1.26673939202022297364452388972, 2.24895683337272173927983718538, 3.02113625063977938342676736027, 4.04967922504011595429373759486, 4.42100420657712233064020254978, 5.68881968275444669872880949719, 6.44474571909013809095829520191, 7.16804590368143171155605018595, 7.64005798105140620305917364890
