| L(s) = 1 | + 2-s + 4-s + 8-s + 11-s + 2·13-s + 16-s − 4·17-s − 6·19-s + 22-s + 4·23-s − 5·25-s + 2·26-s + 10·29-s − 6·31-s + 32-s − 4·34-s − 6·37-s − 6·38-s − 12·41-s − 8·43-s + 44-s + 4·46-s + 2·47-s − 5·50-s + 2·52-s − 6·53-s + 10·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 0.301·11-s + 0.554·13-s + 1/4·16-s − 0.970·17-s − 1.37·19-s + 0.213·22-s + 0.834·23-s − 25-s + 0.392·26-s + 1.85·29-s − 1.07·31-s + 0.176·32-s − 0.685·34-s − 0.986·37-s − 0.973·38-s − 1.87·41-s − 1.21·43-s + 0.150·44-s + 0.589·46-s + 0.291·47-s − 0.707·50-s + 0.277·52-s − 0.824·53-s + 1.31·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.04911294487863644947847596171, −6.54457046719268389329282564708, −6.12928016763066447729285563880, −5.08316559941505302422469262039, −4.64702589496963019502504749433, −3.80292644671654860631338543879, −3.21020151266381282202645511235, −2.18842563100362433161385287823, −1.50131290588153234653626168607, 0,
1.50131290588153234653626168607, 2.18842563100362433161385287823, 3.21020151266381282202645511235, 3.80292644671654860631338543879, 4.64702589496963019502504749433, 5.08316559941505302422469262039, 6.12928016763066447729285563880, 6.54457046719268389329282564708, 7.04911294487863644947847596171
