| L(s) = 1 | − 2-s − 0.484·3-s + 4-s + 1.12·5-s + 0.484·6-s − 8-s − 2.76·9-s − 1.12·10-s + 5.76·11-s − 0.484·12-s − 5.12·13-s − 0.545·15-s + 16-s − 5.60·17-s + 2.76·18-s + 2.96·19-s + 1.12·20-s − 5.76·22-s + 7.28·23-s + 0.484·24-s − 3.73·25-s + 5.12·26-s + 2.79·27-s + 29-s + 0.545·30-s − 4.15·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.279·3-s + 0.5·4-s + 0.503·5-s + 0.197·6-s − 0.353·8-s − 0.921·9-s − 0.355·10-s + 1.73·11-s − 0.139·12-s − 1.42·13-s − 0.140·15-s + 0.250·16-s − 1.36·17-s + 0.651·18-s + 0.681·19-s + 0.251·20-s − 1.22·22-s + 1.51·23-s + 0.0989·24-s − 0.746·25-s + 1.00·26-s + 0.537·27-s + 0.185·29-s + 0.0995·30-s − 0.746·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 0.484T + 3T^{2} \) |
| 5 | \( 1 - 1.12T + 5T^{2} \) |
| 11 | \( 1 - 5.76T + 11T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 + 5.60T + 17T^{2} \) |
| 19 | \( 1 - 2.96T + 19T^{2} \) |
| 23 | \( 1 - 7.28T + 23T^{2} \) |
| 31 | \( 1 + 4.15T + 31T^{2} \) |
| 37 | \( 1 + 2.24T + 37T^{2} \) |
| 41 | \( 1 + 7.67T + 41T^{2} \) |
| 43 | \( 1 - 0.734T + 43T^{2} \) |
| 47 | \( 1 - 1.12T + 47T^{2} \) |
| 53 | \( 1 + 1.76T + 53T^{2} \) |
| 59 | \( 1 + 6.39T + 59T^{2} \) |
| 61 | \( 1 + 2.24T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 + 2.31T + 71T^{2} \) |
| 73 | \( 1 - 2.39T + 73T^{2} \) |
| 79 | \( 1 + 0.545T + 79T^{2} \) |
| 83 | \( 1 - 18.1T + 83T^{2} \) |
| 89 | \( 1 - 1.29T + 89T^{2} \) |
| 97 | \( 1 + 8.16T + 97T^{2} \) |
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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
−8.726517695131243020158474704119, −7.59142498146607782212240267371, −6.85127014858482552994862583854, −6.34424738927309331361241004763, −5.40728885196457659541502847049, −4.61755081597725071514614252993, −3.38649616572374770543238060922, −2.42156525128541122599618301005, −1.42740331496756921108740654946, 0,
1.42740331496756921108740654946, 2.42156525128541122599618301005, 3.38649616572374770543238060922, 4.61755081597725071514614252993, 5.40728885196457659541502847049, 6.34424738927309331361241004763, 6.85127014858482552994862583854, 7.59142498146607782212240267371, 8.726517695131243020158474704119
