| L(s) = 1 | − 4·4-s + 50·9-s + 88·11-s + 16·16-s − 38·19-s − 532·29-s + 272·31-s − 200·36-s + 940·41-s − 352·44-s + 622·49-s − 1.47e3·59-s + 1.30e3·61-s − 64·64-s − 432·71-s + 152·76-s + 2.44e3·79-s + 1.77e3·81-s − 204·89-s + 4.40e3·99-s + 2.43e3·101-s + 2.12e3·116-s + 3.14e3·121-s − 1.08e3·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.85·9-s + 2.41·11-s + 1/4·16-s − 0.458·19-s − 3.40·29-s + 1.57·31-s − 0.925·36-s + 3.58·41-s − 1.20·44-s + 1.81·49-s − 3.24·59-s + 2.72·61-s − 1/8·64-s − 0.722·71-s + 0.229·76-s + 3.47·79-s + 2.42·81-s − 0.242·89-s + 4.46·99-s + 2.39·101-s + 1.70·116-s + 2.36·121-s − 0.787·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.173342871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.173342871\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 50 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 622 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 p T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4350 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 17278 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 266 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 136 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 78470 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 470 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 103318 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 150046 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 296458 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 736 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 650 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 87374 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 216 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 713518 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1220 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 670230 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 102 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 186946 T^{2} + p^{6} T^{4} \) |
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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
−9.488743992559296704330255142240, −9.444178071999462987272701110870, −9.241874048878530021936631144085, −8.908488963585789896691217180728, −8.088182481288461088562007254956, −7.71348763933945680288112917948, −7.27929084209887201084413591490, −7.03000623695892262910975406380, −6.36637029629134043799226019369, −6.11507537596890031272187046696, −5.65260107244294519398559938002, −4.88351742056757147204851747186, −4.21611517652519882864966679529, −4.20547943667275625366310172548, −3.82972217073167767676585243229, −3.21563638554872820601229795143, −2.01365239843043587836641391806, −1.91431057936983102874297653181, −0.853998204788565281250230192921, −0.853128948808835242502800541807,
0.853128948808835242502800541807, 0.853998204788565281250230192921, 1.91431057936983102874297653181, 2.01365239843043587836641391806, 3.21563638554872820601229795143, 3.82972217073167767676585243229, 4.20547943667275625366310172548, 4.21611517652519882864966679529, 4.88351742056757147204851747186, 5.65260107244294519398559938002, 6.11507537596890031272187046696, 6.36637029629134043799226019369, 7.03000623695892262910975406380, 7.27929084209887201084413591490, 7.71348763933945680288112917948, 8.088182481288461088562007254956, 8.908488963585789896691217180728, 9.241874048878530021936631144085, 9.444178071999462987272701110870, 9.488743992559296704330255142240
