| L(s) = 1 | − 8·7-s − 16·13-s + 32·19-s + 32·25-s + 88·31-s + 68·37-s + 80·43-s − 50·49-s − 100·61-s − 16·67-s − 32·73-s − 152·79-s + 128·91-s + 352·97-s − 56·103-s − 46·121-s + 127-s + 131-s − 256·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 146·169-s + ⋯ |
| L(s) = 1 | − 8/7·7-s − 1.23·13-s + 1.68·19-s + 1.27·25-s + 2.83·31-s + 1.83·37-s + 1.86·43-s − 1.02·49-s − 1.63·61-s − 0.238·67-s − 0.438·73-s − 1.92·79-s + 1.40·91-s + 3.62·97-s − 0.543·103-s − 0.380·121-s + 0.00787·127-s + 0.00763·131-s − 1.92·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.863·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.938196506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.938196506\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 32 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )( 1 + 14 T + p^{2} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 416 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 770 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1664 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 44 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 1184 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2782 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4160 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5810 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 7490 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 76 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 334 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15680 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 176 T + p^{2} T^{2} )^{2} \) |
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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
−10.44882329623251988886633529092, −10.35234643651384293253900603358, −9.654189517006674460684717817632, −9.641766286495501295678462943012, −9.129393105507582494888270405018, −8.628668861269995369771418312991, −7.901414971593580332363917520714, −7.65565122356251904222467093466, −7.18317636281495382331816379031, −6.68016659667006142708363365500, −6.05134961276261328388706934089, −5.99403464681577799305453674611, −4.95170075849796652750007623942, −4.78134354405892140433843986155, −4.18765788399385224104826914129, −3.30162842428833956952982956579, −2.71658937507894777929043254397, −2.70000423826547743465227700644, −1.26417550957377167244564087988, −0.58848311151016488524009744380,
0.58848311151016488524009744380, 1.26417550957377167244564087988, 2.70000423826547743465227700644, 2.71658937507894777929043254397, 3.30162842428833956952982956579, 4.18765788399385224104826914129, 4.78134354405892140433843986155, 4.95170075849796652750007623942, 5.99403464681577799305453674611, 6.05134961276261328388706934089, 6.68016659667006142708363365500, 7.18317636281495382331816379031, 7.65565122356251904222467093466, 7.901414971593580332363917520714, 8.628668861269995369771418312991, 9.129393105507582494888270405018, 9.641766286495501295678462943012, 9.654189517006674460684717817632, 10.35234643651384293253900603358, 10.44882329623251988886633529092
