| L(s) = 1 | + 2·5-s − 6·9-s − 2·17-s + 5·25-s + 10·29-s + 2·37-s − 10·41-s − 12·45-s − 14·49-s − 14·53-s + 10·61-s + 6·73-s + 27·81-s − 4·85-s + 20·89-s + 36·97-s + 2·101-s + 12·109-s + 14·113-s − 22·121-s + 22·125-s + 127-s + 131-s + 137-s + 139-s + 20·145-s + 149-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2·9-s − 0.485·17-s + 25-s + 1.85·29-s + 0.328·37-s − 1.56·41-s − 1.78·45-s − 2·49-s − 1.92·53-s + 1.28·61-s + 0.702·73-s + 3·81-s − 0.433·85-s + 2.11·89-s + 3.65·97-s + 0.199·101-s + 1.14·109-s + 1.31·113-s − 2·121-s + 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.66·145-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29246464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29246464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.115441790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.115441790\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−8.339404407806784979805063868463, −8.259940276924542474205691752400, −7.56251640150692743131899460682, −7.44269096659002166816521247633, −6.65470608501995748967457375573, −6.39600320238416325869614619073, −6.24545166216826244940788587369, −6.08506653508983395876665948111, −5.34041642192029349053559932620, −5.06024215683950327731809265843, −4.87616598719280955289602771171, −4.55691729694495304277263072565, −3.66054072176450290699381826535, −3.41617768225560963040881789265, −2.93410897743996099493501531590, −2.74152690375811162388719710257, −1.98012986105323017224554684664, −1.93205456336423180723752762190, −0.969544949121715982708019921159, −0.42824352894671353067376791343,
0.42824352894671353067376791343, 0.969544949121715982708019921159, 1.93205456336423180723752762190, 1.98012986105323017224554684664, 2.74152690375811162388719710257, 2.93410897743996099493501531590, 3.41617768225560963040881789265, 3.66054072176450290699381826535, 4.55691729694495304277263072565, 4.87616598719280955289602771171, 5.06024215683950327731809265843, 5.34041642192029349053559932620, 6.08506653508983395876665948111, 6.24545166216826244940788587369, 6.39600320238416325869614619073, 6.65470608501995748967457375573, 7.44269096659002166816521247633, 7.56251640150692743131899460682, 8.259940276924542474205691752400, 8.339404407806784979805063868463
