| L(s) = 1 | − 4-s − 9-s − 6·11-s + 16-s + 10·19-s + 14·31-s + 36-s + 4·41-s + 6·44-s + 5·49-s + 4·61-s − 64-s + 4·71-s − 10·76-s + 10·79-s + 81-s + 20·89-s + 6·99-s + 14·101-s − 10·109-s + 5·121-s − 14·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 1.80·11-s + 1/4·16-s + 2.29·19-s + 2.51·31-s + 1/6·36-s + 0.624·41-s + 0.904·44-s + 5/7·49-s + 0.512·61-s − 1/8·64-s + 0.474·71-s − 1.14·76-s + 1.12·79-s + 1/9·81-s + 2.11·89-s + 0.603·99-s + 1.39·101-s − 0.957·109-s + 5/11·121-s − 1.25·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.168294228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.168294228\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.063984090501083176599876361603, −8.711810076340587079370358538648, −8.172970612614319067779007494073, −7.994025721984093618081686782111, −7.62880002683200570807575205675, −7.41631173451711677372486432169, −6.72720772258649592744899043674, −6.46549599912781176754928565284, −5.81024039913179164477572307631, −5.47729904627234782254924807134, −5.29907367924317703658930503650, −4.69633561241729804880854703105, −4.55517624917281616818850059403, −3.84319716434560564264625713413, −3.14826157351925821323774196468, −3.08724985745249765195677137774, −2.49832566884053435184392029293, −1.97532343030252417689562816126, −0.910388054158702927031553283718, −0.65194847193031442096880134526,
0.65194847193031442096880134526, 0.910388054158702927031553283718, 1.97532343030252417689562816126, 2.49832566884053435184392029293, 3.08724985745249765195677137774, 3.14826157351925821323774196468, 3.84319716434560564264625713413, 4.55517624917281616818850059403, 4.69633561241729804880854703105, 5.29907367924317703658930503650, 5.47729904627234782254924807134, 5.81024039913179164477572307631, 6.46549599912781176754928565284, 6.72720772258649592744899043674, 7.41631173451711677372486432169, 7.62880002683200570807575205675, 7.994025721984093618081686782111, 8.172970612614319067779007494073, 8.711810076340587079370358538648, 9.063984090501083176599876361603
