| L(s) = 1 | + 8·4-s + 10·5-s − 13·9-s + 48·16-s + 14·19-s + 80·20-s + 75·25-s − 62·31-s − 104·36-s − 146·41-s − 130·45-s + 98·49-s − 74·59-s + 256·64-s − 274·71-s + 112·76-s + 480·80-s + 88·81-s + 140·95-s + 600·100-s − 154·101-s + 374·109-s + 242·121-s − 496·124-s + 500·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 2·4-s + 2·5-s − 1.44·9-s + 3·16-s + 0.736·19-s + 4·20-s + 3·25-s − 2·31-s − 2.88·36-s − 3.56·41-s − 2.88·45-s + 2·49-s − 1.25·59-s + 4·64-s − 3.85·71-s + 1.47·76-s + 6·80-s + 1.08·81-s + 1.47·95-s + 6·100-s − 1.52·101-s + 3.43·109-s + 2·121-s − 4·124-s + 4·125-s + 0.00787·127-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.881888322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.881888322\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 31 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 3 | $C_2^2$ | \( 1 + 13 T^{2} + p^{4} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 158 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 547 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 562 T^{2} + p^{4} T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2707 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 73 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 53 T^{2} + p^{4} T^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 5573 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 37 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 137 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 533 T^{2} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 10027 T^{2} + p^{4} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
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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
−12.93283599575604809038618825197, −12.37171329931969024912052597073, −11.78839029344775892644815073008, −11.58055521222764394060460783623, −10.74492989595146839762919227293, −10.64318334244917401708250928167, −10.06694714704105214500782623860, −9.548546630560527470738815709639, −8.719881625472620974503357579800, −8.547453833247468578363188164906, −7.32523653144919214548486586833, −7.20214974752350346781327159939, −6.41368037736231320231543733945, −5.87634156240696442535637594764, −5.64730460324874894136086312105, −5.08043744295923998107736226009, −3.27568485321104761348004458513, −2.96544132129744774747153897823, −2.06997191046769389802992918427, −1.55994391202019095413108907569,
1.55994391202019095413108907569, 2.06997191046769389802992918427, 2.96544132129744774747153897823, 3.27568485321104761348004458513, 5.08043744295923998107736226009, 5.64730460324874894136086312105, 5.87634156240696442535637594764, 6.41368037736231320231543733945, 7.20214974752350346781327159939, 7.32523653144919214548486586833, 8.547453833247468578363188164906, 8.719881625472620974503357579800, 9.548546630560527470738815709639, 10.06694714704105214500782623860, 10.64318334244917401708250928167, 10.74492989595146839762919227293, 11.58055521222764394060460783623, 11.78839029344775892644815073008, 12.37171329931969024912052597073, 12.93283599575604809038618825197
