L(s) = 1 | − 2·3-s − 2·4-s − 2·5-s + 3·9-s + 4·12-s + 4·15-s + 4·16-s + 12·17-s − 2·19-s + 4·20-s + 3·25-s − 4·27-s − 4·31-s − 6·36-s − 6·45-s − 8·48-s + 14·49-s − 24·51-s + 4·57-s + 24·59-s − 8·60-s + 16·61-s − 8·64-s − 28·67-s − 24·68-s + 24·71-s − 8·73-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 4-s − 0.894·5-s + 9-s + 1.15·12-s + 1.03·15-s + 16-s + 2.91·17-s − 0.458·19-s + 0.894·20-s + 3/5·25-s − 0.769·27-s − 0.718·31-s − 36-s − 0.894·45-s − 1.15·48-s + 2·49-s − 3.36·51-s + 0.529·57-s + 3.12·59-s − 1.03·60-s + 2.04·61-s − 64-s − 3.42·67-s − 2.91·68-s + 2.84·71-s − 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1299600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1299600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9815914649\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9815914649\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.00284776619157623584949678480, −9.952753702058317235486458428325, −9.064755388368542347218925361600, −8.896028357859668382757650361408, −8.268997100707044686258612308921, −8.017273508294965188159976937507, −7.40537943139300794819143384849, −7.28282910237281797083821412931, −6.78009636990962524842319607063, −5.89987687363066667710503038131, −5.74583906057244411219055954539, −5.39330811232755232228694813402, −4.95809856600611204500962269368, −4.33207742999256182938578336075, −3.97589852102795396988357662847, −3.50634288456804243985694459244, −3.07642288341028889216107407967, −1.99189113851944057619718018579, −0.969457217288431726792467462037, −0.65341728497806199044911160030,
0.65341728497806199044911160030, 0.969457217288431726792467462037, 1.99189113851944057619718018579, 3.07642288341028889216107407967, 3.50634288456804243985694459244, 3.97589852102795396988357662847, 4.33207742999256182938578336075, 4.95809856600611204500962269368, 5.39330811232755232228694813402, 5.74583906057244411219055954539, 5.89987687363066667710503038131, 6.78009636990962524842319607063, 7.28282910237281797083821412931, 7.40537943139300794819143384849, 8.017273508294965188159976937507, 8.268997100707044686258612308921, 8.896028357859668382757650361408, 9.064755388368542347218925361600, 9.952753702058317235486458428325, 10.00284776619157623584949678480