| L(s) = 1 | − 4·2-s + 18·3-s + 8·4-s − 30·5-s − 72·6-s + 58·7-s + 162·9-s + 120·10-s − 236·11-s + 144·12-s + 138·13-s − 232·14-s − 540·15-s − 64·16-s − 542·17-s − 648·18-s − 240·20-s + 1.04e3·21-s + 944·22-s + 538·23-s + 275·25-s − 552·26-s + 1.45e3·27-s + 464·28-s + 2.16e3·30-s + 404·31-s + 256·32-s + ⋯ |
| L(s) = 1 | − 2-s + 2·3-s + 1/2·4-s − 6/5·5-s − 2·6-s + 1.18·7-s + 2·9-s + 6/5·10-s − 1.95·11-s + 12-s + 0.816·13-s − 1.18·14-s − 2.39·15-s − 1/4·16-s − 1.87·17-s − 2·18-s − 3/5·20-s + 2.36·21-s + 1.95·22-s + 1.01·23-s + 0.439·25-s − 0.816·26-s + 2·27-s + 0.591·28-s + 12/5·30-s + 0.420·31-s + 1/4·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.000457096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.000457096\) |
| \(L(3)\) |
|
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 | $C_2$ | \( 1 + p^{2} T + p^{3} T^{2} \) |
| 5 | $C_2$ | \( 1 + 6 p T + p^{4} T^{2} \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{4} T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 58 T + 1682 T^{2} - 58 p^{4} T^{3} + p^{8} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 118 T + p^{4} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 138 T + 9522 T^{2} - 138 p^{4} T^{3} + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 542 T + 146882 T^{2} + 542 p^{4} T^{3} + p^{8} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 182242 T^{2} + p^{8} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 538 T + 144722 T^{2} - 538 p^{4} T^{3} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 952162 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 202 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 1302 T + 847602 T^{2} + 1302 p^{4} T^{3} + p^{8} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 1682 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 2178 T + 2371842 T^{2} - 2178 p^{4} T^{3} + p^{8} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 54 p T + 1458 p^{2} T^{2} - 54 p^{5} T^{3} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 1222 T + 746642 T^{2} + 1222 p^{4} T^{3} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 22889122 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 5598 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 1502 T + 1128002 T^{2} + 1502 p^{4} T^{3} + p^{8} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6442 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 5902 T + 17416802 T^{2} + 5902 p^{4} T^{3} + p^{8} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 33613438 T^{2} + p^{8} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12462 T + 77650722 T^{2} + 12462 p^{4} T^{3} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 84185918 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 14622 T + 106901442 T^{2} + 14622 p^{4} T^{3} + p^{8} T^{4} \) |
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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
−20.40927505534607014480708864434, −19.83766310813537809024435622667, −19.37004415030012598391470815480, −18.57683806843483705103387378519, −18.17528192393846041571962560345, −17.39657263892531068990599874961, −15.92584702397339250974258949644, −15.58359723051238049094321415300, −15.14653380220326052994219103775, −14.09689758628053902298710611143, −13.50900885747239028179362041665, −12.56296324811360941388841825951, −10.92683241127515175220492978638, −10.84596160869241973199237549649, −9.104051683033048217187035624946, −8.576419223140330273316556010163, −7.961397833103900235480438224705, −7.41236298354186449641074172392, −4.46846646297317084744423291953, −2.63299884198847838915051193532,
2.63299884198847838915051193532, 4.46846646297317084744423291953, 7.41236298354186449641074172392, 7.961397833103900235480438224705, 8.576419223140330273316556010163, 9.104051683033048217187035624946, 10.84596160869241973199237549649, 10.92683241127515175220492978638, 12.56296324811360941388841825951, 13.50900885747239028179362041665, 14.09689758628053902298710611143, 15.14653380220326052994219103775, 15.58359723051238049094321415300, 15.92584702397339250974258949644, 17.39657263892531068990599874961, 18.17528192393846041571962560345, 18.57683806843483705103387378519, 19.37004415030012598391470815480, 19.83766310813537809024435622667, 20.40927505534607014480708864434
