L(s) = 1 | − 3·5-s + 9-s − 13-s − 12·17-s + 4·25-s + 9·29-s + 9·37-s − 2·41-s − 3·45-s − 5·49-s + 2·53-s + 3·61-s + 3·65-s − 12·73-s + 81-s + 36·85-s + 7·89-s + 28·97-s + 21·101-s + 20·109-s + 10·113-s − 117-s − 8·121-s + 3·125-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | − 1.34·5-s + 1/3·9-s − 0.277·13-s − 2.91·17-s + 4/5·25-s + 1.67·29-s + 1.47·37-s − 0.312·41-s − 0.447·45-s − 5/7·49-s + 0.274·53-s + 0.384·61-s + 0.372·65-s − 1.40·73-s + 1/9·81-s + 3.90·85-s + 0.741·89-s + 2.84·97-s + 2.08·101-s + 1.91·109-s + 0.940·113-s − 0.0924·117-s − 0.727·121-s + 0.268·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
good | 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
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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
−7.948408827896996543815906824209, −7.43589504455609081830175717924, −7.17139393847728310694001067832, −6.64903529837760500208688538122, −6.28422415881554531460263977563, −5.88849714863554153398934478412, −4.78699757266908977633138956017, −4.60047740744136897498299198527, −4.52583853023470297076143931686, −3.71670221672702036806514497172, −3.28781363925697437220364601416, −2.45392062355583278521702760109, −2.11992308220281432654830919826, −0.922000152810741677470887942473, 0,
0.922000152810741677470887942473, 2.11992308220281432654830919826, 2.45392062355583278521702760109, 3.28781363925697437220364601416, 3.71670221672702036806514497172, 4.52583853023470297076143931686, 4.60047740744136897498299198527, 4.78699757266908977633138956017, 5.88849714863554153398934478412, 6.28422415881554531460263977563, 6.64903529837760500208688538122, 7.17139393847728310694001067832, 7.43589504455609081830175717924, 7.948408827896996543815906824209