| L(s) = 1 | + 2·9-s − 2·11-s + 6·29-s − 20·31-s − 20·41-s + 20·59-s + 20·61-s + 6·71-s − 26·79-s − 5·81-s − 20·89-s − 4·99-s − 20·101-s − 26·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 0.603·11-s + 1.11·29-s − 3.59·31-s − 3.12·41-s + 2.60·59-s + 2.56·61-s + 0.712·71-s − 2.92·79-s − 5/9·81-s − 2.11·89-s − 0.402·99-s − 1.99·101-s − 2.49·109-s − 1.72·121-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 + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6202690705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6202690705\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 49 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−8.479780481918265994114772981061, −8.396919150111542506385287243097, −7.57315257699442388065483324722, −7.30070816748168728670874731436, −7.06341236442745217485343518751, −6.60687228164933156603825049216, −6.58761326160545569996550848008, −5.63928315415110487195383417100, −5.50132617461632731064280111727, −5.16947121174559560350480550373, −5.00850408712891194826939959623, −4.08837257644761070241174294629, −4.00657304398252802592430232357, −3.68359321194566141461283702519, −3.08604894330988074959902592730, −2.59047335427488914792070474290, −2.19955543502290383046998659232, −1.51409863939534245663434146025, −1.32395086792341645603887287922, −0.20949553402196277461448617906,
0.20949553402196277461448617906, 1.32395086792341645603887287922, 1.51409863939534245663434146025, 2.19955543502290383046998659232, 2.59047335427488914792070474290, 3.08604894330988074959902592730, 3.68359321194566141461283702519, 4.00657304398252802592430232357, 4.08837257644761070241174294629, 5.00850408712891194826939959623, 5.16947121174559560350480550373, 5.50132617461632731064280111727, 5.63928315415110487195383417100, 6.58761326160545569996550848008, 6.60687228164933156603825049216, 7.06341236442745217485343518751, 7.30070816748168728670874731436, 7.57315257699442388065483324722, 8.396919150111542506385287243097, 8.479780481918265994114772981061
