| L(s) = 1 | − 2-s + 4-s − 8-s + 3·11-s − 2·13-s + 16-s − 3·17-s + 19-s − 3·22-s + 6·23-s − 5·25-s + 2·26-s − 6·29-s + 4·31-s − 32-s + 3·34-s − 4·37-s − 38-s + 9·41-s − 43-s + 3·44-s − 6·46-s − 6·47-s + 5·50-s − 2·52-s − 12·53-s + 6·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.904·11-s − 0.554·13-s + 1/4·16-s − 0.727·17-s + 0.229·19-s − 0.639·22-s + 1.25·23-s − 25-s + 0.392·26-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.514·34-s − 0.657·37-s − 0.162·38-s + 1.40·41-s − 0.152·43-s + 0.452·44-s − 0.884·46-s − 0.875·47-s + 0.707·50-s − 0.277·52-s − 1.64·53-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−7.57486075433264375877702850521, −6.83856700403258930718083224666, −6.35782135459828133998327355434, −5.48525873198846016080664036461, −4.67165130826644550690102729752, −3.84523654364808887700815920458, −2.98455636915823987690855594247, −2.07379971708251484770168742942, −1.22338111669750545882567189127, 0,
1.22338111669750545882567189127, 2.07379971708251484770168742942, 2.98455636915823987690855594247, 3.84523654364808887700815920458, 4.67165130826644550690102729752, 5.48525873198846016080664036461, 6.35782135459828133998327355434, 6.83856700403258930718083224666, 7.57486075433264375877702850521
