| L(s) = 1 | − 4·4-s + 53·9-s − 70·11-s + 16·16-s − 46·19-s + 348·29-s + 152·31-s − 212·36-s + 862·41-s + 280·44-s − 49·49-s − 152·59-s + 236·61-s − 64·64-s + 1.06e3·71-s + 184·76-s − 804·79-s + 2.08e3·81-s + 1.59e3·89-s − 3.71e3·99-s − 4.31e3·109-s − 1.39e3·116-s + 1.01e3·121-s − 608·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.96·9-s − 1.91·11-s + 1/4·16-s − 0.555·19-s + 2.22·29-s + 0.880·31-s − 0.981·36-s + 3.28·41-s + 0.959·44-s − 1/7·49-s − 0.335·59-s + 0.495·61-s − 1/8·64-s + 1.77·71-s + 0.277·76-s − 1.14·79-s + 2.85·81-s + 1.90·89-s − 3.76·99-s − 3.79·109-s − 1.11·116-s + 0.761·121-s − 0.440·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.365567058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.365567058\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 53 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 35 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1030 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 1623 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 23 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 15666 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 p T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 76 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 67450 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 431 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 138278 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 69030 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 286090 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 76 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 118 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 129557 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 530 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 688633 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 402 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 338965 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 799 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 454754 T^{2} + p^{6} T^{4} \) |
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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
−10.89421870974176361465853790197, −10.84212807115090456503884134878, −10.32416403757690434196821766029, −9.973328770981539789662527552841, −9.574740993641261696030500873511, −9.062620568347734434397155098211, −8.326240774117670352668189774469, −7.973325348386137593655553935944, −7.62980574955028382158072853685, −7.08676926267681557439343556390, −6.45063664809193521523657224026, −6.05326093741876727290094840664, −5.06630318954005728149838732402, −4.93363110495113528438243397222, −4.25004878076648885642715814187, −3.88726582201250126346357293246, −2.71405124806053664274780317018, −2.44996613213504002805825963995, −1.26791827763781424213911891655, −0.61039865183274919285378898341,
0.61039865183274919285378898341, 1.26791827763781424213911891655, 2.44996613213504002805825963995, 2.71405124806053664274780317018, 3.88726582201250126346357293246, 4.25004878076648885642715814187, 4.93363110495113528438243397222, 5.06630318954005728149838732402, 6.05326093741876727290094840664, 6.45063664809193521523657224026, 7.08676926267681557439343556390, 7.62980574955028382158072853685, 7.973325348386137593655553935944, 8.326240774117670352668189774469, 9.062620568347734434397155098211, 9.574740993641261696030500873511, 9.973328770981539789662527552841, 10.32416403757690434196821766029, 10.84212807115090456503884134878, 10.89421870974176361465853790197
