| L(s) = 1 | − 6·5-s + 4·7-s + 12·17-s + 17·25-s − 24·35-s + 2·37-s + 6·41-s + 20·43-s + 12·47-s + 9·49-s − 12·59-s + 4·67-s + 28·79-s − 12·83-s − 72·85-s − 18·89-s − 36·101-s − 22·109-s + 48·119-s − 5·121-s − 18·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2.68·5-s + 1.51·7-s + 2.91·17-s + 17/5·25-s − 4.05·35-s + 0.328·37-s + 0.937·41-s + 3.04·43-s + 1.75·47-s + 9/7·49-s − 1.56·59-s + 0.488·67-s + 3.15·79-s − 1.31·83-s − 7.80·85-s − 1.90·89-s − 3.58·101-s − 2.10·109-s + 4.40·119-s − 0.454·121-s − 1.60·125-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.542462125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.542462125\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 115 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−10.89805773830269221677894138933, −10.32662390470099114042626281047, −9.492523271290072175519272352245, −9.387314706235804928673910862984, −8.610287254908329453313484316998, −8.044381298194251625091481078884, −7.961556206862797752154927028510, −7.79811924854856239466814564671, −7.22069651647561925637587961467, −7.10425072879305392438658208391, −5.90273728814475041929538409788, −5.67158657434365688064545595987, −5.10732015917325321888303997167, −4.45338876063546516600267605152, −4.04746434361632659910029501738, −3.86416325348673531727637195774, −3.12451173773084013834398890970, −2.55425424424961638468067319166, −1.28279118866956139487243444151, −0.75941191870987508666180872782,
0.75941191870987508666180872782, 1.28279118866956139487243444151, 2.55425424424961638468067319166, 3.12451173773084013834398890970, 3.86416325348673531727637195774, 4.04746434361632659910029501738, 4.45338876063546516600267605152, 5.10732015917325321888303997167, 5.67158657434365688064545595987, 5.90273728814475041929538409788, 7.10425072879305392438658208391, 7.22069651647561925637587961467, 7.79811924854856239466814564671, 7.961556206862797752154927028510, 8.044381298194251625091481078884, 8.610287254908329453313484316998, 9.387314706235804928673910862984, 9.492523271290072175519272352245, 10.32662390470099114042626281047, 10.89805773830269221677894138933
