| L(s) = 1 | − 1.73·2-s + 0.999·4-s + 2.18·5-s + 0.791·7-s + 1.73·8-s − 3.79·10-s − 0.456·11-s − 5.79·13-s − 1.37·14-s − 5·16-s − 0.913·17-s − 5.58·19-s + 2.18·20-s + 0.791·22-s − 0.208·25-s + 10.0·26-s + 0.791·28-s − 4.83·29-s − 5.58·31-s + 5.19·32-s + 1.58·34-s + 1.73·35-s + 1.79·37-s + 9.66·38-s + 3.79·40-s + 6.20·41-s + 5.37·43-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.499·4-s + 0.978·5-s + 0.299·7-s + 0.612·8-s − 1.19·10-s − 0.137·11-s − 1.60·13-s − 0.366·14-s − 1.25·16-s − 0.221·17-s − 1.28·19-s + 0.489·20-s + 0.168·22-s − 0.0417·25-s + 1.96·26-s + 0.149·28-s − 0.897·29-s − 1.00·31-s + 0.918·32-s + 0.271·34-s + 0.292·35-s + 0.294·37-s + 1.56·38-s + 0.599·40-s + 0.969·41-s + 0.819·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 5 | \( 1 - 2.18T + 5T^{2} \) |
| 7 | \( 1 - 0.791T + 7T^{2} \) |
| 11 | \( 1 + 0.456T + 11T^{2} \) |
| 13 | \( 1 + 5.79T + 13T^{2} \) |
| 17 | \( 1 + 0.913T + 17T^{2} \) |
| 19 | \( 1 + 5.58T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 4.83T + 29T^{2} \) |
| 31 | \( 1 + 5.58T + 31T^{2} \) |
| 37 | \( 1 - 1.79T + 37T^{2} \) |
| 41 | \( 1 - 6.20T + 41T^{2} \) |
| 43 | \( 1 - 5.37T + 43T^{2} \) |
| 47 | \( 1 + 2.18T + 47T^{2} \) |
| 53 | \( 1 + 4.83T + 53T^{2} \) |
| 59 | \( 1 - 1.82T + 59T^{2} \) |
| 61 | \( 1 + 2.79T + 61T^{2} \) |
| 67 | \( 1 + 5.79T + 67T^{2} \) |
| 71 | \( 1 - 8.29T + 71T^{2} \) |
| 73 | \( 1 - 0.373T + 73T^{2} \) |
| 79 | \( 1 + 3.58T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 - 0.361T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−9.468837668595862312090504332253, −8.784584680005138360369520018905, −7.81355268358225698559003953468, −7.21716517869590579173435674631, −6.16997625067518496300684597713, −5.15262923818704157036797946471, −4.26583424595497426360933894620, −2.45385609489656468781674335054, −1.74118548818167295784471943582, 0,
1.74118548818167295784471943582, 2.45385609489656468781674335054, 4.26583424595497426360933894620, 5.15262923818704157036797946471, 6.16997625067518496300684597713, 7.21716517869590579173435674631, 7.81355268358225698559003953468, 8.784584680005138360369520018905, 9.468837668595862312090504332253
