| L(s) = 1 | + 4·7-s − 9-s − 16·17-s − 16·23-s − 25-s − 16·31-s + 12·41-s − 8·47-s − 2·49-s − 4·63-s + 16·71-s − 12·73-s − 16·79-s + 81-s − 12·89-s + 20·97-s − 28·103-s + 8·113-s − 64·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 16·153-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1/3·9-s − 3.88·17-s − 3.33·23-s − 1/5·25-s − 2.87·31-s + 1.87·41-s − 1.16·47-s − 2/7·49-s − 0.503·63-s + 1.89·71-s − 1.40·73-s − 1.80·79-s + 1/9·81-s − 1.27·89-s + 2.03·97-s − 2.75·103-s + 0.752·113-s − 5.86·119-s + 1.63·121-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 + 1.29·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3722746673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3722746673\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−9.554343354705391074190311498462, −8.922273054198033453709846062297, −8.480816447226991951363825020543, −8.357979313340841615406334446328, −7.937611105732909857402969901764, −7.42216095749336299088482879341, −7.18450772301760934505289873997, −6.47625507190133155543061480174, −6.34872993418641526197923181299, −5.71045300169062296344142850572, −5.50416124649113896993274680609, −4.78269026619529346254278463336, −4.43565291227071627820194002960, −4.12803910061574219114783887546, −3.85798986198352850602846487532, −2.95141362301300755999034324486, −2.19015664243196136746358153714, −1.85657974064969429133039585456, −1.83715760987286641735144717654, −0.20081749695667809393906003040,
0.20081749695667809393906003040, 1.83715760987286641735144717654, 1.85657974064969429133039585456, 2.19015664243196136746358153714, 2.95141362301300755999034324486, 3.85798986198352850602846487532, 4.12803910061574219114783887546, 4.43565291227071627820194002960, 4.78269026619529346254278463336, 5.50416124649113896993274680609, 5.71045300169062296344142850572, 6.34872993418641526197923181299, 6.47625507190133155543061480174, 7.18450772301760934505289873997, 7.42216095749336299088482879341, 7.937611105732909857402969901764, 8.357979313340841615406334446328, 8.480816447226991951363825020543, 8.922273054198033453709846062297, 9.554343354705391074190311498462
