L(s) = 1 | + 4·4-s + 6·11-s + 12·16-s − 5·25-s − 18·29-s + 24·44-s + 32·64-s + 24·71-s + 2·79-s − 20·100-s + 22·109-s − 72·116-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 19·169-s + 173-s + 72·176-s + 179-s + 181-s + ⋯ |
L(s) = 1 | + 2·4-s + 1.80·11-s + 3·16-s − 25-s − 3.34·29-s + 3.61·44-s + 4·64-s + 2.84·71-s + 0.225·79-s − 2·100-s + 2.10·109-s − 6.68·116-s + 5/11·121-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 − 1.46·169-s + 0.0760·173-s + 5.42·176-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.087587895\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.087587895\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | | \( 1 \) |
good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 149 T^{2} + p^{2} T^{4} \) |
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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
−9.371059841556066167526026663404, −8.894906836104855504452077993308, −8.436855195587168631105189545732, −7.85151571939304969434131569200, −7.56975380788859788257358982809, −7.39471658529375921098455680408, −6.84833226466404512755412142014, −6.40891496103094300770859251850, −6.40204357860456518017262622576, −5.71675920082583197683234924659, −5.54097090215634869573652336963, −5.03260973859624312324749565359, −4.07497607912174528141005543020, −3.84103101574617743385933003172, −3.54834642237086968305671203142, −2.98832972993189075799072485191, −2.27649644787652347374740204513, −1.79660858322491714909986436321, −1.66710869575317664509841209561, −0.75247110204299960144919557136,
0.75247110204299960144919557136, 1.66710869575317664509841209561, 1.79660858322491714909986436321, 2.27649644787652347374740204513, 2.98832972993189075799072485191, 3.54834642237086968305671203142, 3.84103101574617743385933003172, 4.07497607912174528141005543020, 5.03260973859624312324749565359, 5.54097090215634869573652336963, 5.71675920082583197683234924659, 6.40204357860456518017262622576, 6.40891496103094300770859251850, 6.84833226466404512755412142014, 7.39471658529375921098455680408, 7.56975380788859788257358982809, 7.85151571939304969434131569200, 8.436855195587168631105189545732, 8.894906836104855504452077993308, 9.371059841556066167526026663404