| L(s) = 1 | − 4-s − 9-s + 16-s − 4·19-s − 18·29-s − 20·31-s + 36-s + 24·41-s + 13·49-s + 12·59-s + 28·61-s − 64-s + 6·71-s + 4·76-s − 16·79-s + 81-s − 6·89-s − 18·101-s − 22·109-s + 18·116-s − 22·121-s + 20·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 1/4·16-s − 0.917·19-s − 3.34·29-s − 3.59·31-s + 1/6·36-s + 3.74·41-s + 13/7·49-s + 1.56·59-s + 3.58·61-s − 1/8·64-s + 0.712·71-s + 0.458·76-s − 1.80·79-s + 1/9·81-s − 0.635·89-s − 1.79·101-s − 2.10·109-s + 1.67·116-s − 2·121-s + 1.79·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8275536970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8275536970\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.718237525568172304344904483514, −8.605051765250741409100546314634, −7.962259243959860198193963418122, −7.64932742460707046542319130297, −7.34147782862468235489264738559, −6.98347171447754643438213838256, −6.71776481873990189119981420432, −5.83232504593998188376952984720, −5.68629807227761166986993999394, −5.39800810465250056963474902579, −5.36206780451406251723517096260, −4.19488962960997314416677710871, −4.16364353677489225796839222990, −3.82065648193455679506069218430, −3.50989211916932776681356517836, −2.61188766267494037976637182749, −2.25458655580470329699796194227, −1.91755864360957928797763209637, −1.09368263445562024420625367736, −0.29830048807557271985178934611,
0.29830048807557271985178934611, 1.09368263445562024420625367736, 1.91755864360957928797763209637, 2.25458655580470329699796194227, 2.61188766267494037976637182749, 3.50989211916932776681356517836, 3.82065648193455679506069218430, 4.16364353677489225796839222990, 4.19488962960997314416677710871, 5.36206780451406251723517096260, 5.39800810465250056963474902579, 5.68629807227761166986993999394, 5.83232504593998188376952984720, 6.71776481873990189119981420432, 6.98347171447754643438213838256, 7.34147782862468235489264738559, 7.64932742460707046542319130297, 7.962259243959860198193963418122, 8.605051765250741409100546314634, 8.718237525568172304344904483514
