| L(s) = 1 | − 8·7-s + 10·25-s + 8·31-s + 34·49-s + 20·73-s + 8·79-s + 28·97-s + 40·103-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s − 80·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 2·25-s + 1.43·31-s + 34/7·49-s + 2.34·73-s + 0.900·79-s + 2.84·97-s + 3.94·103-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s − 6.04·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.474585992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.474585992\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−9.109355025347877427749698838547, −9.008428359699540435016744051413, −8.554947232483548683002580658808, −8.124169989877371713769418839823, −7.35968717154217309943927251970, −7.35928195643356245391222192985, −6.66926198963039532733097882902, −6.44909564911114125700282328747, −6.26610341501198060980853030440, −5.94757960444598584508574878041, −5.06280644591501156631301521343, −5.02021312284690983688973493731, −4.29872888615737020177104784318, −3.74383113409750442137554660928, −3.29021543040717627143857293621, −3.14661758904341400413257763534, −2.59733218933344662632698734592, −2.11261587435062822607495516605, −0.907548119678773926813708962442, −0.54054744680117114954700093718,
0.54054744680117114954700093718, 0.907548119678773926813708962442, 2.11261587435062822607495516605, 2.59733218933344662632698734592, 3.14661758904341400413257763534, 3.29021543040717627143857293621, 3.74383113409750442137554660928, 4.29872888615737020177104784318, 5.02021312284690983688973493731, 5.06280644591501156631301521343, 5.94757960444598584508574878041, 6.26610341501198060980853030440, 6.44909564911114125700282328747, 6.66926198963039532733097882902, 7.35928195643356245391222192985, 7.35968717154217309943927251970, 8.124169989877371713769418839823, 8.554947232483548683002580658808, 9.008428359699540435016744051413, 9.109355025347877427749698838547
