| L(s) = 1 | − 4-s − 4·7-s + 6·13-s − 3·16-s + 6·19-s + 25-s + 4·28-s + 2·31-s − 8·37-s − 10·43-s + 2·49-s − 6·52-s + 4·61-s + 7·64-s − 14·67-s − 6·76-s + 4·79-s − 24·91-s + 2·97-s − 100-s + 28·103-s − 24·109-s + 12·112-s − 4·121-s − 2·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.51·7-s + 1.66·13-s − 3/4·16-s + 1.37·19-s + 1/5·25-s + 0.755·28-s + 0.359·31-s − 1.31·37-s − 1.52·43-s + 2/7·49-s − 0.832·52-s + 0.512·61-s + 7/8·64-s − 1.71·67-s − 0.688·76-s + 0.450·79-s − 2.51·91-s + 0.203·97-s − 0.0999·100-s + 2.75·103-s − 2.29·109-s + 1.13·112-s − 0.363·121-s − 0.179·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.255360423462715209168018903284, −7.53475293787070780905295708732, −7.17181001068942514933126862250, −6.55837165840372060726346851759, −6.38338293152944269940879376425, −5.93171982463535917019017484152, −5.22033215255672620850904885032, −4.98598676245323315788210901850, −4.17106231067447187066318367227, −3.70382491597435883966281116650, −3.24923691385426228788415144395, −2.92157462896746393067989153495, −1.87831161997820156325930422244, −1.06581621034428724597756516792, 0,
1.06581621034428724597756516792, 1.87831161997820156325930422244, 2.92157462896746393067989153495, 3.24923691385426228788415144395, 3.70382491597435883966281116650, 4.17106231067447187066318367227, 4.98598676245323315788210901850, 5.22033215255672620850904885032, 5.93171982463535917019017484152, 6.38338293152944269940879376425, 6.55837165840372060726346851759, 7.17181001068942514933126862250, 7.53475293787070780905295708732, 8.255360423462715209168018903284
