L(s) = 1 | − 2-s + 5-s + 8-s − 9-s − 10-s − 2·13-s − 16-s + 17-s + 18-s + 25-s + 2·26-s − 2·29-s − 34-s − 37-s + 40-s + 41-s − 45-s − 49-s − 50-s − 2·53-s + 2·58-s + 61-s + 64-s − 2·65-s − 72-s + 4·73-s + 74-s + ⋯ |
L(s) = 1 | − 2-s + 5-s + 8-s − 9-s − 10-s − 2·13-s − 16-s + 17-s + 18-s + 25-s + 2·26-s − 2·29-s − 34-s − 37-s + 40-s + 41-s − 45-s − 49-s − 50-s − 2·53-s + 2·58-s + 61-s + 64-s − 2·65-s − 72-s + 4·73-s + 74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2597781081\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2597781081\) |
\(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 37 | $C_2$ | \( 1 + T + T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$ | \( ( 1 - T )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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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
−13.85653146219689087176981720472, −12.75470100440307614342689893959, −12.71611033927127344535869253417, −12.10175890340696933277954386344, −11.21638905910060143805761389051, −11.03020978873252149269118182461, −10.32016329323130379036524398995, −9.742048422211040471347849514437, −9.426467174471067737455659293965, −9.266654306864399411552743721958, −8.287180582741234828554201407768, −7.962051909607678993465956431210, −7.32941209753467813664766886872, −6.77605692322991098383087074077, −5.91882122147529498271655254947, −5.11379352929463618926966154777, −5.07555161507183373009572892502, −3.75374885505549034519191665355, −2.70594643519740103407472915111, −1.86391576307011732057411614853,
1.86391576307011732057411614853, 2.70594643519740103407472915111, 3.75374885505549034519191665355, 5.07555161507183373009572892502, 5.11379352929463618926966154777, 5.91882122147529498271655254947, 6.77605692322991098383087074077, 7.32941209753467813664766886872, 7.962051909607678993465956431210, 8.287180582741234828554201407768, 9.266654306864399411552743721958, 9.426467174471067737455659293965, 9.742048422211040471347849514437, 10.32016329323130379036524398995, 11.03020978873252149269118182461, 11.21638905910060143805761389051, 12.10175890340696933277954386344, 12.71611033927127344535869253417, 12.75470100440307614342689893959, 13.85653146219689087176981720472