| L(s) = 1 | − 2·3-s − 5·5-s − 2·7-s + 3·9-s + 2·11-s − 13-s + 10·15-s − 10·17-s − 2·19-s + 4·21-s − 2·23-s + 10·25-s − 4·27-s − 8·29-s + 4·31-s − 4·33-s + 10·35-s − 6·37-s + 2·39-s − 16·41-s + 3·43-s − 15·45-s + 6·47-s + 3·49-s + 20·51-s + 3·53-s − 10·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2.23·5-s − 0.755·7-s + 9-s + 0.603·11-s − 0.277·13-s + 2.58·15-s − 2.42·17-s − 0.458·19-s + 0.872·21-s − 0.417·23-s + 2·25-s − 0.769·27-s − 1.48·29-s + 0.718·31-s − 0.696·33-s + 1.69·35-s − 0.986·37-s + 0.320·39-s − 2.49·41-s + 0.457·43-s − 2.23·45-s + 0.875·47-s + 3/7·49-s + 2.80·51-s + 0.412·53-s − 1.34·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59721984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59721984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1993365359\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1993365359\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_4$ | \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + T - 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 69 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 63 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 141 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 27 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 107 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 5 T + 123 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 27 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 73 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 7 T + 93 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 151 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 31 T + 417 T^{2} + 31 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 20 T + 289 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
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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
−8.015672636755053911095907203718, −7.56979351190679437081300223937, −7.21123290024469583069396201299, −6.95115601489987768791253286869, −6.69017447137998912765507837661, −6.56439706633810364214912151376, −5.84920794602332160283740966383, −5.78955159859682254445254249875, −5.01418921158176147548637499573, −4.94989970894876461359380500093, −4.27491581951200184371741147893, −4.16797382316433369939937273686, −3.78252671957082252916395746561, −3.72111112863357672990138547861, −2.94830031917551792198713365863, −2.56295710337596195342554985602, −1.86929475973830976230283202296, −1.52090931534457295814492378661, −0.47940065137436031845109445852, −0.24110398745860507173760918972,
0.24110398745860507173760918972, 0.47940065137436031845109445852, 1.52090931534457295814492378661, 1.86929475973830976230283202296, 2.56295710337596195342554985602, 2.94830031917551792198713365863, 3.72111112863357672990138547861, 3.78252671957082252916395746561, 4.16797382316433369939937273686, 4.27491581951200184371741147893, 4.94989970894876461359380500093, 5.01418921158176147548637499573, 5.78955159859682254445254249875, 5.84920794602332160283740966383, 6.56439706633810364214912151376, 6.69017447137998912765507837661, 6.95115601489987768791253286869, 7.21123290024469583069396201299, 7.56979351190679437081300223937, 8.015672636755053911095907203718
