| L(s) = 1 | + 0.414·2-s + 0.414·3-s − 1.82·4-s + 5-s + 0.171·6-s − 1.58·8-s − 2.82·9-s + 0.414·10-s + 2.41·11-s − 0.757·12-s − 1.82·13-s + 0.414·15-s + 3·16-s + 4.82·17-s − 1.17·18-s − 6·19-s − 1.82·20-s + 0.999·22-s − 7.65·23-s − 0.656·24-s − 4·25-s − 0.757·26-s − 2.41·27-s + 29-s + 0.171·30-s + 4.07·31-s + 4.41·32-s + ⋯ |
| L(s) = 1 | + 0.292·2-s + 0.239·3-s − 0.914·4-s + 0.447·5-s + 0.0700·6-s − 0.560·8-s − 0.942·9-s + 0.130·10-s + 0.727·11-s − 0.218·12-s − 0.507·13-s + 0.106·15-s + 0.750·16-s + 1.17·17-s − 0.276·18-s − 1.37·19-s − 0.408·20-s + 0.213·22-s − 1.59·23-s − 0.134·24-s − 0.800·25-s − 0.148·26-s − 0.464·27-s + 0.185·29-s + 0.0313·30-s + 0.731·31-s + 0.780·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1421 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1421 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 3 | \( 1 - 0.414T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 13 | \( 1 + 1.82T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 31 | \( 1 - 4.07T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 - 6.41T + 43T^{2} \) |
| 47 | \( 1 + 5.24T + 47T^{2} \) |
| 53 | \( 1 + 7.48T + 53T^{2} \) |
| 59 | \( 1 + 7.65T + 59T^{2} \) |
| 61 | \( 1 + 0.828T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 + 3.17T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 0.414T + 79T^{2} \) |
| 83 | \( 1 - 3.65T + 83T^{2} \) |
| 89 | \( 1 + 4.48T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−9.135163131769828451322502846733, −8.389859758107560087934312768846, −7.80754705654854972585542693747, −6.35196305622295694607093414268, −5.86088651056772820143980441275, −4.90620688117245662609971275561, −3.97753786431012311894244708926, −3.11606352339714025864212296710, −1.82651671575356652048232957226, 0,
1.82651671575356652048232957226, 3.11606352339714025864212296710, 3.97753786431012311894244708926, 4.90620688117245662609971275561, 5.86088651056772820143980441275, 6.35196305622295694607093414268, 7.80754705654854972585542693747, 8.389859758107560087934312768846, 9.135163131769828451322502846733
