| L(s) = 1 | + 3·4-s − 2·5-s + 5·16-s − 12·19-s − 6·20-s − 25-s + 14·29-s + 4·31-s − 10·41-s + 20·59-s + 14·61-s + 3·64-s + 4·71-s − 36·76-s + 4·79-s − 10·80-s + 18·89-s + 24·95-s − 3·100-s − 18·101-s − 10·109-s + 42·116-s − 22·121-s + 12·124-s + 12·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 0.894·5-s + 5/4·16-s − 2.75·19-s − 1.34·20-s − 1/5·25-s + 2.59·29-s + 0.718·31-s − 1.56·41-s + 2.60·59-s + 1.79·61-s + 3/8·64-s + 0.474·71-s − 4.12·76-s + 0.450·79-s − 1.11·80-s + 1.90·89-s + 2.46·95-s − 0.299·100-s − 1.79·101-s − 0.957·109-s + 3.89·116-s − 2·121-s + 1.07·124-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-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
| \(L(1)\) |
\(\approx\) |
\(2.366474785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.366474785\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−9.005721950386104510193050956034, −8.781463012032600775405514612103, −8.236919575136484304759945259970, −8.031665869146663784360190212584, −8.002593780853915334196442492114, −7.08865788918300873212201085017, −6.76163196756106836001498723552, −6.64846475464119245159452323063, −6.43522401972856488515551031478, −5.78893371095455507012794998173, −5.34901866502769447346510679810, −4.67063800747134133910320644145, −4.47932906901295203631781553988, −3.73938194793370249089260598056, −3.66137531674170346067645569998, −2.77883224633474499668418361798, −2.43530946453746251375592528062, −2.13092699896002722083603516910, −1.34771243201509797842344860427, −0.52519361875167666678867616061,
0.52519361875167666678867616061, 1.34771243201509797842344860427, 2.13092699896002722083603516910, 2.43530946453746251375592528062, 2.77883224633474499668418361798, 3.66137531674170346067645569998, 3.73938194793370249089260598056, 4.47932906901295203631781553988, 4.67063800747134133910320644145, 5.34901866502769447346510679810, 5.78893371095455507012794998173, 6.43522401972856488515551031478, 6.64846475464119245159452323063, 6.76163196756106836001498723552, 7.08865788918300873212201085017, 8.002593780853915334196442492114, 8.031665869146663784360190212584, 8.236919575136484304759945259970, 8.781463012032600775405514612103, 9.005721950386104510193050956034
