| L(s) = 1 | − 3-s + 4-s − 4·7-s + 9-s − 12-s + 16-s + 2·19-s + 4·21-s + 25-s − 27-s − 4·28-s + 36-s + 8·37-s − 12·43-s − 48-s − 2·49-s − 2·57-s + 4·61-s − 4·63-s + 64-s − 4·73-s − 75-s + 2·76-s − 16·79-s + 81-s + 4·84-s + 16·97-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 1.51·7-s + 1/3·9-s − 0.288·12-s + 1/4·16-s + 0.458·19-s + 0.872·21-s + 1/5·25-s − 0.192·27-s − 0.755·28-s + 1/6·36-s + 1.31·37-s − 1.82·43-s − 0.144·48-s − 2/7·49-s − 0.264·57-s + 0.512·61-s − 0.503·63-s + 1/8·64-s − 0.468·73-s − 0.115·75-s + 0.229·76-s − 1.80·79-s + 1/9·81-s + 0.436·84-s + 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 974700 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 974700 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−7.65735578435109364022918175994, −7.47231387261999183818040048050, −6.92060556397032542472367252370, −6.56383114270620232058265921883, −6.07657152725821217624879228381, −5.98634616587230595166705000030, −5.22611296970313731007175215014, −4.80411785408799003074467608381, −4.24244004877095440204684144895, −3.45191506153810335863912261323, −3.27946907924628736034072094506, −2.63124068493624222532667688726, −1.89151962432729389992611169280, −1.02054089713911743443106437818, 0,
1.02054089713911743443106437818, 1.89151962432729389992611169280, 2.63124068493624222532667688726, 3.27946907924628736034072094506, 3.45191506153810335863912261323, 4.24244004877095440204684144895, 4.80411785408799003074467608381, 5.22611296970313731007175215014, 5.98634616587230595166705000030, 6.07657152725821217624879228381, 6.56383114270620232058265921883, 6.92060556397032542472367252370, 7.47231387261999183818040048050, 7.65735578435109364022918175994
