| L(s) = 1 | − 2·5-s + 4·11-s − 4·19-s − 25-s + 14·29-s − 12·31-s + 10·41-s + 13·49-s − 8·55-s + 24·59-s − 14·61-s − 20·71-s − 8·79-s + 30·89-s + 8·95-s + 36·101-s + 18·109-s − 10·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s − 28·145-s + 149-s + 151-s + 24·155-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.20·11-s − 0.917·19-s − 1/5·25-s + 2.59·29-s − 2.15·31-s + 1.56·41-s + 13/7·49-s − 1.07·55-s + 3.12·59-s − 1.79·61-s − 2.37·71-s − 0.900·79-s + 3.17·89-s + 0.820·95-s + 3.58·101-s + 1.72·109-s − 0.909·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.32·145-s + 0.0819·149-s + 0.0813·151-s + 1.92·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.360537683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.360537683\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 141 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 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
−8.688697550528906421703855723760, −8.681219298728717122003589845683, −8.106324933181246073271535891830, −7.63908301867832351629288871846, −7.36708139987818038339714872553, −7.07711825669967354336044817363, −6.59139920759231274892026945146, −6.28039624078053686812626798589, −5.75484881076922668375601454728, −5.63931910241735834882935539007, −4.77855225979936868094672047421, −4.54686052722220136967113920059, −4.09481503051361900507466022365, −3.92407203689559373377488182881, −3.26151904876089954673208357998, −3.00250387261997784144413529995, −2.10543357778851679743634363863, −1.97419906298211168993612599406, −0.955520220371236636751676724623, −0.59271170599321519961571695133,
0.59271170599321519961571695133, 0.955520220371236636751676724623, 1.97419906298211168993612599406, 2.10543357778851679743634363863, 3.00250387261997784144413529995, 3.26151904876089954673208357998, 3.92407203689559373377488182881, 4.09481503051361900507466022365, 4.54686052722220136967113920059, 4.77855225979936868094672047421, 5.63931910241735834882935539007, 5.75484881076922668375601454728, 6.28039624078053686812626798589, 6.59139920759231274892026945146, 7.07711825669967354336044817363, 7.36708139987818038339714872553, 7.63908301867832351629288871846, 8.106324933181246073271535891830, 8.681219298728717122003589845683, 8.688697550528906421703855723760
