| L(s) = 1 | + 2·7-s + 6·9-s − 12·17-s + 6·25-s + 16·31-s − 4·41-s − 16·47-s + 3·49-s + 12·63-s + 16·71-s − 20·73-s + 32·79-s + 27·81-s + 12·89-s − 12·97-s + 32·103-s + 4·113-s − 24·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 72·153-s + 157-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 2·9-s − 2.91·17-s + 6/5·25-s + 2.87·31-s − 0.624·41-s − 2.33·47-s + 3/7·49-s + 1.51·63-s + 1.89·71-s − 2.34·73-s + 3.60·79-s + 3·81-s + 1.27·89-s − 1.21·97-s + 3.15·103-s + 0.376·113-s − 2.20·119-s + 6/11·121-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 − 5.82·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.059339015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.059339015\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−9.455564899697810592407247663845, −9.105361001425330309830059408909, −8.542412218189265106204449080903, −8.431007374193734365174550384293, −7.919749478658128228399276073598, −7.51503733376692892887126903313, −6.95009324022285998257289448568, −6.74282655436372727131412527977, −6.34838798950487487519238798343, −6.22118238633745572350094155708, −4.98023886211419191598620813075, −4.86400637365457365779841486142, −4.70044314759144490956577086640, −4.24523287725253576647584269256, −3.72728598893114482228370967194, −3.08945340157153625565987591156, −2.30882421588956529155062429466, −2.04685660685479934351914453189, −1.37384435999096037927494630262, −0.69607963127277611925559392862,
0.69607963127277611925559392862, 1.37384435999096037927494630262, 2.04685660685479934351914453189, 2.30882421588956529155062429466, 3.08945340157153625565987591156, 3.72728598893114482228370967194, 4.24523287725253576647584269256, 4.70044314759144490956577086640, 4.86400637365457365779841486142, 4.98023886211419191598620813075, 6.22118238633745572350094155708, 6.34838798950487487519238798343, 6.74282655436372727131412527977, 6.95009324022285998257289448568, 7.51503733376692892887126903313, 7.919749478658128228399276073598, 8.431007374193734365174550384293, 8.542412218189265106204449080903, 9.105361001425330309830059408909, 9.455564899697810592407247663845
