L(s) = 1 | + 48·7-s + 86·9-s − 2.39e3·17-s − 368·23-s + 774·25-s − 1.14e4·31-s + 1.77e4·41-s + 4.73e4·47-s − 3.18e4·49-s + 4.12e3·63-s + 6.39e4·71-s + 9.77e3·73-s + 8.91e4·79-s − 5.16e4·81-s − 1.43e5·89-s + 9.77e4·97-s − 3.60e5·103-s − 4.70e4·113-s − 1.15e5·119-s + 3.06e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2.06e5·153-s + ⋯ |
L(s) = 1 | + 0.370·7-s + 0.353·9-s − 2.01·17-s − 0.145·23-s + 0.247·25-s − 2.14·31-s + 1.65·41-s + 3.12·47-s − 1.89·49-s + 0.131·63-s + 1.50·71-s + 0.214·73-s + 1.60·79-s − 0.874·81-s − 1.92·89-s + 1.05·97-s − 3.35·103-s − 0.346·113-s − 0.744·119-s + 1.90·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s − 0.711·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.661718034\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.661718034\) |
\(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 86 T^{2} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 774 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 24 T + p^{5} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 306726 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 514102 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 1198 T + p^{5} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 4313738 T^{2} + p^{10} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 p T + p^{5} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 30250774 T^{2} + p^{10} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 5728 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 32061638 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8886 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 209597542 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 23664 T + p^{5} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 699828390 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 1145049222 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 1347608278 T^{2} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2459007190 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 31960 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4886 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 44560 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 3340172790 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 71994 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 48866 T + p^{5} T^{2} )^{2} \) |
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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
−11.20893585873722462824573429754, −10.87806424959077785702887401027, −10.75040589700776725337575406126, −9.935374803085159455557329184629, −9.229950268671409782653893734179, −9.169575027248295091103858572276, −8.589217787170878939399746916915, −7.86954116099159367580833754256, −7.55129662951040587956018011145, −6.72148416912916828270731567983, −6.69912533323737357094511685650, −5.67053095283426978958064181116, −5.40172816013318895153980493445, −4.45689683287032976796443187614, −4.23253079952235268324931161162, −3.53163498938008397447214813871, −2.51397833256581579649624963905, −2.11416570911757065959179987682, −1.29168142364925529930606681074, −0.37179498207568367946466172821,
0.37179498207568367946466172821, 1.29168142364925529930606681074, 2.11416570911757065959179987682, 2.51397833256581579649624963905, 3.53163498938008397447214813871, 4.23253079952235268324931161162, 4.45689683287032976796443187614, 5.40172816013318895153980493445, 5.67053095283426978958064181116, 6.69912533323737357094511685650, 6.72148416912916828270731567983, 7.55129662951040587956018011145, 7.86954116099159367580833754256, 8.589217787170878939399746916915, 9.169575027248295091103858572276, 9.229950268671409782653893734179, 9.935374803085159455557329184629, 10.75040589700776725337575406126, 10.87806424959077785702887401027, 11.20893585873722462824573429754