| L(s) = 1 | + 2·3-s − 5·5-s + 20·7-s + 27·9-s + 69·11-s + 22·13-s − 10·15-s + 102·17-s − 47·19-s + 40·21-s + 51·23-s + 154·27-s + 444·29-s − 152·31-s + 138·33-s − 100·35-s − 221·37-s + 44·39-s − 666·41-s + 580·43-s − 135·45-s + 555·47-s + 57·49-s + 204·51-s − 261·53-s − 345·55-s − 94·57-s + ⋯ |
| L(s) = 1 | + 0.384·3-s − 0.447·5-s + 1.07·7-s + 9-s + 1.89·11-s + 0.469·13-s − 0.172·15-s + 1.45·17-s − 0.567·19-s + 0.415·21-s + 0.462·23-s + 1.09·27-s + 2.84·29-s − 0.880·31-s + 0.727·33-s − 0.482·35-s − 0.981·37-s + 0.180·39-s − 2.53·41-s + 2.05·43-s − 0.447·45-s + 1.72·47-s + 0.166·49-s + 0.560·51-s − 0.676·53-s − 0.845·55-s − 0.218·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.711521486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.711521486\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 - 20 T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T - 23 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 69 T + 3430 T^{2} - 69 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 11 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 p T + 19 p^{2} T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 47 T - 4650 T^{2} + 47 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 51 T - 9566 T^{2} - 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 222 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 152 T - 6687 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 221 T - 1812 T^{2} + 221 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 333 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 290 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 555 T + 204202 T^{2} - 555 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 261 T - 80756 T^{2} + 261 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 360 T - 75779 T^{2} + 360 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 824 T + 451995 T^{2} + 824 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 776 T + 301413 T^{2} + 776 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 720 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 580 T - 52617 T^{2} - 580 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 226 T - 441963 T^{2} - 226 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 816 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 918 T + 137755 T^{2} + 918 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 470 T + p^{3} T^{2} )^{2} \) |
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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
−12.58146468305415016397593961621, −12.49944914840292719232249202828, −11.88875170324046149935912110940, −11.65854567791371046078829257108, −10.77836585413201515786673477677, −10.44583586836662776476608098072, −9.945028067009851069643293665728, −9.129616272315894144073397537857, −8.577069677460821126834766056257, −8.499490914528388408400523867791, −7.36982520899840053099050186850, −7.34008063875117775405754008779, −6.44978243038684495243359120076, −5.91385745772034069597634008255, −4.75432428777268948975243830807, −4.49291269886912340231220240231, −3.71829397426889419586091097460, −2.99968601133589616232173511165, −1.50033479052297356923907658363, −1.20040061418756743223296399477,
1.20040061418756743223296399477, 1.50033479052297356923907658363, 2.99968601133589616232173511165, 3.71829397426889419586091097460, 4.49291269886912340231220240231, 4.75432428777268948975243830807, 5.91385745772034069597634008255, 6.44978243038684495243359120076, 7.34008063875117775405754008779, 7.36982520899840053099050186850, 8.499490914528388408400523867791, 8.577069677460821126834766056257, 9.129616272315894144073397537857, 9.945028067009851069643293665728, 10.44583586836662776476608098072, 10.77836585413201515786673477677, 11.65854567791371046078829257108, 11.88875170324046149935912110940, 12.49944914840292719232249202828, 12.58146468305415016397593961621
