| L(s) = 1 | + 208·17-s − 234·25-s − 944·41-s − 686·49-s + 2.19e3·73-s − 352·89-s + 1.18e3·97-s + 2.65e3·113-s + 2.66e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4.07e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 2.96·17-s − 1.87·25-s − 3.59·41-s − 2·49-s + 3.52·73-s − 0.419·89-s + 1.24·97-s + 2.21·113-s + 2·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 1.85·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.240405862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.240405862\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p^{3} T^{2} )( 1 + 4 T + p^{3} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 18 T + p^{3} T^{2} )( 1 + 18 T + p^{3} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 104 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 284 T + p^{3} T^{2} )( 1 + 284 T + p^{3} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 214 T + p^{3} T^{2} )( 1 + 214 T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 472 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 572 T + p^{3} T^{2} )( 1 + 572 T + p^{3} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 830 T + p^{3} T^{2} )( 1 + 830 T + p^{3} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 1098 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 176 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 594 T + p^{3} T^{2} )^{2} \) |
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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
−9.732953960746515101223919517471, −9.459575567983006801211560838401, −8.694770171897172745619544128182, −8.153446041973150164978631847715, −8.123635703419419196953461610066, −7.71420663486240630148559518068, −7.08276952125274223408447709788, −6.86029747532526183396515388647, −6.03161652659051841864626223642, −5.98153055837876300370539954559, −5.29994290890271469750969986721, −5.04436048771782344601069793970, −4.54563533469717558309558860576, −3.55210932264918301656539141648, −3.50755159231955498694552543596, −3.21580376133176961370314166557, −2.14246599443990054505614360189, −1.75864231969248505801397874591, −1.11774815787704034665666049057, −0.38698956432979135456994068457,
0.38698956432979135456994068457, 1.11774815787704034665666049057, 1.75864231969248505801397874591, 2.14246599443990054505614360189, 3.21580376133176961370314166557, 3.50755159231955498694552543596, 3.55210932264918301656539141648, 4.54563533469717558309558860576, 5.04436048771782344601069793970, 5.29994290890271469750969986721, 5.98153055837876300370539954559, 6.03161652659051841864626223642, 6.86029747532526183396515388647, 7.08276952125274223408447709788, 7.71420663486240630148559518068, 8.123635703419419196953461610066, 8.153446041973150164978631847715, 8.694770171897172745619544128182, 9.459575567983006801211560838401, 9.732953960746515101223919517471
