| L(s) = 1 | − 3-s − 2·4-s + 5-s − 2·9-s + 2·12-s + 2·13-s − 15-s + 4·16-s − 2·20-s + 25-s + 5·27-s − 2·31-s + 4·36-s − 4·37-s − 2·39-s − 15·41-s + 2·43-s − 2·45-s − 4·48-s + 5·49-s − 4·52-s + 2·60-s − 8·64-s + 2·65-s + 17·67-s − 12·71-s − 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.447·5-s − 2/3·9-s + 0.577·12-s + 0.554·13-s − 0.258·15-s + 16-s − 0.447·20-s + 1/5·25-s + 0.962·27-s − 0.359·31-s + 2/3·36-s − 0.657·37-s − 0.320·39-s − 2.34·41-s + 0.304·43-s − 0.298·45-s − 0.577·48-s + 5/7·49-s − 0.554·52-s + 0.258·60-s − 64-s + 0.248·65-s + 2.07·67-s − 1.42·71-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.442612976295692083072812045929, −7.72586393772078563816270675964, −7.25594125353782219176008997932, −6.59179077875367505603847367346, −6.33539806031813324770540675955, −5.72151650607909997817089162098, −5.33427479438455035808461112683, −5.05753015659967150055924252211, −4.48683361446820090556200679677, −3.76006413532330039492022263365, −3.43380189559290797214183065627, −2.71592421994019696595250435769, −1.86527795912980428737770388057, −1.02571346765272912354389153886, 0,
1.02571346765272912354389153886, 1.86527795912980428737770388057, 2.71592421994019696595250435769, 3.43380189559290797214183065627, 3.76006413532330039492022263365, 4.48683361446820090556200679677, 5.05753015659967150055924252211, 5.33427479438455035808461112683, 5.72151650607909997817089162098, 6.33539806031813324770540675955, 6.59179077875367505603847367346, 7.25594125353782219176008997932, 7.72586393772078563816270675964, 8.442612976295692083072812045929
