| L(s) = 1 | + 3-s − 2.82·5-s + 1.41·7-s + 9-s − 0.585·11-s − 13-s − 2.82·15-s − 2.82·17-s − 5.41·19-s + 1.41·21-s + 2.82·23-s + 3.00·25-s + 27-s + 3.65·29-s + 5.41·31-s − 0.585·33-s − 4.00·35-s + 2·37-s − 39-s + 6.82·41-s + 5.65·43-s − 2.82·45-s + 3.41·47-s − 5·49-s − 2.82·51-s + 1.17·53-s + 1.65·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.26·5-s + 0.534·7-s + 0.333·9-s − 0.176·11-s − 0.277·13-s − 0.730·15-s − 0.685·17-s − 1.24·19-s + 0.308·21-s + 0.589·23-s + 0.600·25-s + 0.192·27-s + 0.679·29-s + 0.972·31-s − 0.101·33-s − 0.676·35-s + 0.328·37-s − 0.160·39-s + 1.06·41-s + 0.862·43-s − 0.421·45-s + 0.498·47-s − 0.714·49-s − 0.396·51-s + 0.160·53-s + 0.223·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 + 0.585T + 11T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + 5.41T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 - 5.41T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 6.82T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 - 3.41T + 47T^{2} \) |
| 53 | \( 1 - 1.17T + 53T^{2} \) |
| 59 | \( 1 - 2.24T + 59T^{2} \) |
| 61 | \( 1 + 9.65T + 61T^{2} \) |
| 67 | \( 1 + 3.75T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 7.65T + 73T^{2} \) |
| 79 | \( 1 - 2.82T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 - 2.82T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.45401159877643472333430580938, −6.81993559246700951052758124237, −6.08141493042775272216604633389, −4.99141904963258433337613437068, −4.32896725224365640373133022564, −4.02931828445420780147704500080, −2.91807369012842310362861919265, −2.37368184676150373629102747256, −1.18367910643351639352337399373, 0,
1.18367910643351639352337399373, 2.37368184676150373629102747256, 2.91807369012842310362861919265, 4.02931828445420780147704500080, 4.32896725224365640373133022564, 4.99141904963258433337613437068, 6.08141493042775272216604633389, 6.81993559246700951052758124237, 7.45401159877643472333430580938
