L(s) = 1 | + 2-s + 4-s + 8-s − 6·11-s − 5·13-s + 16-s + 6·17-s + 4·19-s − 6·22-s − 6·23-s − 5·25-s − 5·26-s − 6·29-s + 31-s + 32-s + 6·34-s − 37-s + 4·38-s − 6·41-s − 43-s − 6·44-s − 6·46-s − 6·47-s − 5·50-s − 5·52-s + 6·53-s − 6·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.80·11-s − 1.38·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 1.27·22-s − 1.25·23-s − 25-s − 0.980·26-s − 1.11·29-s + 0.179·31-s + 0.176·32-s + 1.02·34-s − 0.164·37-s + 0.648·38-s − 0.937·41-s − 0.152·43-s − 0.904·44-s − 0.884·46-s − 0.875·47-s − 0.707·50-s − 0.693·52-s + 0.824·53-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−8.018373233993352526075351795604, −7.76500604494403870217253478491, −7.08399389581964304580937840750, −5.76516762621587586677817121204, −5.43294840515298920841807654421, −4.67577382905622175502731044741, −3.54405748811653734427126185430, −2.77713768583350957199647900568, −1.85393606295065766044693801451, 0,
1.85393606295065766044693801451, 2.77713768583350957199647900568, 3.54405748811653734427126185430, 4.67577382905622175502731044741, 5.43294840515298920841807654421, 5.76516762621587586677817121204, 7.08399389581964304580937840750, 7.76500604494403870217253478491, 8.018373233993352526075351795604