| L(s) = 1 | − 2-s − 2.70·3-s + 4-s + 1.80·5-s + 2.70·6-s − 1.02·7-s − 8-s + 4.33·9-s − 1.80·10-s − 11-s − 2.70·12-s − 3.68·13-s + 1.02·14-s − 4.90·15-s + 16-s + 0.383·17-s − 4.33·18-s + 1.48·19-s + 1.80·20-s + 2.78·21-s + 22-s + 7.06·23-s + 2.70·24-s − 1.72·25-s + 3.68·26-s − 3.62·27-s − 1.02·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.56·3-s + 0.5·4-s + 0.809·5-s + 1.10·6-s − 0.388·7-s − 0.353·8-s + 1.44·9-s − 0.572·10-s − 0.301·11-s − 0.781·12-s − 1.02·13-s + 0.274·14-s − 1.26·15-s + 0.250·16-s + 0.0931·17-s − 1.02·18-s + 0.339·19-s + 0.404·20-s + 0.607·21-s + 0.213·22-s + 1.47·23-s + 0.552·24-s − 0.344·25-s + 0.722·26-s − 0.697·27-s − 0.194·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 + 2.70T + 3T^{2} \) |
| 5 | \( 1 - 1.80T + 5T^{2} \) |
| 7 | \( 1 + 1.02T + 7T^{2} \) |
| 13 | \( 1 + 3.68T + 13T^{2} \) |
| 17 | \( 1 - 0.383T + 17T^{2} \) |
| 19 | \( 1 - 1.48T + 19T^{2} \) |
| 23 | \( 1 - 7.06T + 23T^{2} \) |
| 29 | \( 1 + 4.21T + 29T^{2} \) |
| 31 | \( 1 - 0.673T + 31T^{2} \) |
| 37 | \( 1 - 2.90T + 37T^{2} \) |
| 41 | \( 1 + 1.05T + 41T^{2} \) |
| 43 | \( 1 - 9.11T + 43T^{2} \) |
| 47 | \( 1 + 7.00T + 47T^{2} \) |
| 53 | \( 1 - 4.18T + 53T^{2} \) |
| 59 | \( 1 + 8.91T + 59T^{2} \) |
| 61 | \( 1 + 6.12T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + 0.171T + 71T^{2} \) |
| 73 | \( 1 + 7.55T + 73T^{2} \) |
| 79 | \( 1 - 6.71T + 79T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 + 0.122T + 89T^{2} \) |
| 97 | \( 1 - 6.44T + 97T^{2} \) |
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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
−7.79207864970687239601546205290, −7.19510824209419286235165145819, −6.49481562941710392901332736204, −5.87274666855015916875740213190, −5.25957283884856768470479964137, −4.59766722032477715133114977172, −3.18631950388260272780185416485, −2.17146002392186036911131488286, −1.07180326671291083816868305840, 0,
1.07180326671291083816868305840, 2.17146002392186036911131488286, 3.18631950388260272780185416485, 4.59766722032477715133114977172, 5.25957283884856768470479964137, 5.87274666855015916875740213190, 6.49481562941710392901332736204, 7.19510824209419286235165145819, 7.79207864970687239601546205290
