| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s + 2·7-s + 4·8-s − 4·10-s − 2·11-s + 2·13-s + 4·14-s + 5·16-s − 2·17-s + 8·19-s − 6·20-s − 4·22-s + 6·23-s + 3·25-s + 4·26-s + 6·28-s − 4·29-s + 8·31-s + 6·32-s − 4·34-s − 4·35-s − 2·37-s + 16·38-s − 8·40-s − 4·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.755·7-s + 1.41·8-s − 1.26·10-s − 0.603·11-s + 0.554·13-s + 1.06·14-s + 5/4·16-s − 0.485·17-s + 1.83·19-s − 1.34·20-s − 0.852·22-s + 1.25·23-s + 3/5·25-s + 0.784·26-s + 1.13·28-s − 0.742·29-s + 1.43·31-s + 1.06·32-s − 0.685·34-s − 0.676·35-s − 0.328·37-s + 2.59·38-s − 1.26·40-s − 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48024900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48024900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.23336325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.23336325\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 13 | $D_{4}$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 126 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 226 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.914512274229050495382198903191, −7.75011189524773815080149054096, −7.38277780844422831690926755468, −7.04770537802949067762182993537, −6.62017062678692905289358685678, −6.54310921831661339438175862641, −5.75864847131751879590436521974, −5.61323166799819620248324237171, −5.11601226313032921379557658844, −5.03237215294816000542080387542, −4.45145099155886214723814103413, −4.40426093580134160755209163698, −3.58978985418995631824254397101, −3.58070139960632683607869858621, −2.98662528701520603866651991003, −2.86498404271618926306434128665, −2.01516156729491672624801330662, −1.87147261135357698737566112171, −0.888460645267886963845295811357, −0.78457194623569784087579954743,
0.78457194623569784087579954743, 0.888460645267886963845295811357, 1.87147261135357698737566112171, 2.01516156729491672624801330662, 2.86498404271618926306434128665, 2.98662528701520603866651991003, 3.58070139960632683607869858621, 3.58978985418995631824254397101, 4.40426093580134160755209163698, 4.45145099155886214723814103413, 5.03237215294816000542080387542, 5.11601226313032921379557658844, 5.61323166799819620248324237171, 5.75864847131751879590436521974, 6.54310921831661339438175862641, 6.62017062678692905289358685678, 7.04770537802949067762182993537, 7.38277780844422831690926755468, 7.75011189524773815080149054096, 7.914512274229050495382198903191
