| L(s) = 1 | − 2·2-s + 3·3-s + 4-s − 6·6-s − 4·7-s + 2·8-s + 4·9-s + 3·12-s + 8·14-s − 3·16-s + 6·17-s − 8·18-s − 2·19-s − 12·21-s + 6·24-s − 2·25-s − 4·28-s + 2·29-s − 2·32-s − 12·34-s + 4·36-s + 4·38-s + 24·42-s + 6·47-s − 9·48-s + 2·49-s + 4·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.73·3-s + 1/2·4-s − 2.44·6-s − 1.51·7-s + 0.707·8-s + 4/3·9-s + 0.866·12-s + 2.13·14-s − 3/4·16-s + 1.45·17-s − 1.88·18-s − 0.458·19-s − 2.61·21-s + 1.22·24-s − 2/5·25-s − 0.755·28-s + 0.371·29-s − 0.353·32-s − 2.05·34-s + 2/3·36-s + 0.648·38-s + 3.70·42-s + 0.875·47-s − 1.29·48-s + 2/7·49-s + 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1253094 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1253094 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 457 | $C_2$ | \( 1 - 22 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 92 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 118 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
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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
−8.013249683021033775520133618684, −7.50119518100234100838726714640, −7.30498629609357345331632356250, −6.64534315509337861347942657208, −6.30376489914538854892049018821, −5.59678901956651271723340946670, −5.25097200651879468605282362500, −4.32330418728440853277568568057, −3.85998957143927731364061400737, −3.47767561518065872332911381815, −2.95301187213536620750634563643, −2.53333217852887633302004883129, −1.78859555153250873020518882379, −1.03449815567224986964993686027, 0,
1.03449815567224986964993686027, 1.78859555153250873020518882379, 2.53333217852887633302004883129, 2.95301187213536620750634563643, 3.47767561518065872332911381815, 3.85998957143927731364061400737, 4.32330418728440853277568568057, 5.25097200651879468605282362500, 5.59678901956651271723340946670, 6.30376489914538854892049018821, 6.64534315509337861347942657208, 7.30498629609357345331632356250, 7.50119518100234100838726714640, 8.013249683021033775520133618684
