| L(s) = 1 | − 2.90·3-s + 5-s + 0.903·7-s + 5.42·9-s + 1.52·11-s − 0.622·13-s − 2.90·15-s − 7.95·17-s + 1.09·19-s − 2.62·21-s − 7.52·23-s + 25-s − 7.05·27-s − 29-s + 6.90·31-s − 4.42·33-s + 0.903·35-s + 3.95·37-s + 1.80·39-s + 3.67·41-s + 10.5·43-s + 5.42·45-s − 6.90·47-s − 6.18·49-s + 23.0·51-s + 6.42·53-s + 1.52·55-s + ⋯ |
| L(s) = 1 | − 1.67·3-s + 0.447·5-s + 0.341·7-s + 1.80·9-s + 0.459·11-s − 0.172·13-s − 0.749·15-s − 1.92·17-s + 0.251·19-s − 0.572·21-s − 1.56·23-s + 0.200·25-s − 1.35·27-s − 0.185·29-s + 1.23·31-s − 0.770·33-s + 0.152·35-s + 0.650·37-s + 0.289·39-s + 0.573·41-s + 1.60·43-s + 0.809·45-s − 1.00·47-s − 0.883·49-s + 3.23·51-s + 0.883·53-s + 0.205·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 2.90T + 3T^{2} \) |
| 7 | \( 1 - 0.903T + 7T^{2} \) |
| 11 | \( 1 - 1.52T + 11T^{2} \) |
| 13 | \( 1 + 0.622T + 13T^{2} \) |
| 17 | \( 1 + 7.95T + 17T^{2} \) |
| 19 | \( 1 - 1.09T + 19T^{2} \) |
| 23 | \( 1 + 7.52T + 23T^{2} \) |
| 31 | \( 1 - 6.90T + 31T^{2} \) |
| 37 | \( 1 - 3.95T + 37T^{2} \) |
| 41 | \( 1 - 3.67T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 6.90T + 47T^{2} \) |
| 53 | \( 1 - 6.42T + 53T^{2} \) |
| 59 | \( 1 - 1.67T + 59T^{2} \) |
| 61 | \( 1 + 1.86T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 9.13T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 7.80T + 89T^{2} \) |
| 97 | \( 1 + 4.08T + 97T^{2} \) |
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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
−8.691228929031095948933958895420, −7.66507324989410495835813437719, −6.74566487931614398194963253647, −6.20615141065405095154781385533, −5.62802496321630759575767822679, −4.57508898086561690810429420236, −4.22950582484733885682477022627, −2.45410787754701637042230364294, −1.34756792108083830012320212142, 0,
1.34756792108083830012320212142, 2.45410787754701637042230364294, 4.22950582484733885682477022627, 4.57508898086561690810429420236, 5.62802496321630759575767822679, 6.20615141065405095154781385533, 6.74566487931614398194963253647, 7.66507324989410495835813437719, 8.691228929031095948933958895420
