| L(s) = 1 | − 2.44·2-s + 2.04·3-s + 4.00·4-s − 0.489·5-s − 5.00·6-s − 2.06·7-s − 4.90·8-s + 1.16·9-s + 1.19·10-s + 11-s + 8.16·12-s − 4.04·13-s + 5.05·14-s − 15-s + 4.00·16-s − 0.510·17-s − 2.86·18-s − 6.39·19-s − 1.96·20-s − 4.20·21-s − 2.44·22-s + 2.07·23-s − 10.0·24-s − 4.76·25-s + 9.90·26-s − 3.73·27-s − 8.25·28-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 1.17·3-s + 2.00·4-s − 0.219·5-s − 2.04·6-s − 0.779·7-s − 1.73·8-s + 0.389·9-s + 0.379·10-s + 0.301·11-s + 2.35·12-s − 1.12·13-s + 1.35·14-s − 0.258·15-s + 1.00·16-s − 0.123·17-s − 0.674·18-s − 1.46·19-s − 0.438·20-s − 0.918·21-s − 0.522·22-s + 0.433·23-s − 2.04·24-s − 0.952·25-s + 1.94·26-s − 0.719·27-s − 1.55·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 3 | \( 1 - 2.04T + 3T^{2} \) |
| 5 | \( 1 + 0.489T + 5T^{2} \) |
| 7 | \( 1 + 2.06T + 7T^{2} \) |
| 13 | \( 1 + 4.04T + 13T^{2} \) |
| 17 | \( 1 + 0.510T + 17T^{2} \) |
| 19 | \( 1 + 6.39T + 19T^{2} \) |
| 23 | \( 1 - 2.07T + 23T^{2} \) |
| 29 | \( 1 + 2.48T + 29T^{2} \) |
| 31 | \( 1 + 5.23T + 31T^{2} \) |
| 37 | \( 1 - 4.45T + 37T^{2} \) |
| 41 | \( 1 + 0.256T + 41T^{2} \) |
| 43 | \( 1 - 5.76T + 43T^{2} \) |
| 47 | \( 1 - 5.45T + 47T^{2} \) |
| 53 | \( 1 - 5.63T + 53T^{2} \) |
| 59 | \( 1 + 6.29T + 59T^{2} \) |
| 67 | \( 1 + 9.16T + 67T^{2} \) |
| 71 | \( 1 + 1.68T + 71T^{2} \) |
| 73 | \( 1 - 2.67T + 73T^{2} \) |
| 79 | \( 1 + 1.76T + 79T^{2} \) |
| 83 | \( 1 + 9.03T + 83T^{2} \) |
| 89 | \( 1 + 4.40T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−9.679378857573575860983837438035, −9.207996138726079461845485803741, −8.542544557362621367771672824529, −7.67246525321192477665297043097, −7.09931858672790856260801570492, −6.01614542834553048334039879602, −4.10910739803291499959014413306, −2.82813204279153685511136258919, −1.99449765795089920808176596516, 0,
1.99449765795089920808176596516, 2.82813204279153685511136258919, 4.10910739803291499959014413306, 6.01614542834553048334039879602, 7.09931858672790856260801570492, 7.67246525321192477665297043097, 8.542544557362621367771672824529, 9.207996138726079461845485803741, 9.679378857573575860983837438035
