| L(s) = 1 | + 2·3-s + 4-s − 2·7-s + 3·9-s + 8·11-s + 2·12-s − 3·16-s + 10·17-s + 10·19-s − 4·21-s − 2·23-s + 4·27-s − 2·28-s − 4·31-s + 16·33-s + 3·36-s − 4·41-s − 2·43-s + 8·44-s + 8·47-s − 6·48-s − 6·49-s + 20·51-s + 6·53-s + 20·57-s + 8·59-s − 6·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s − 0.755·7-s + 9-s + 2.41·11-s + 0.577·12-s − 3/4·16-s + 2.42·17-s + 2.29·19-s − 0.872·21-s − 0.417·23-s + 0.769·27-s − 0.377·28-s − 0.718·31-s + 2.78·33-s + 1/2·36-s − 0.624·41-s − 0.304·43-s + 1.20·44-s + 1.16·47-s − 0.866·48-s − 6/7·49-s + 2.80·51-s + 0.824·53-s + 2.64·57-s + 1.04·59-s − 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2975625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2975625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.808789323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.808789323\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 122 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 174 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.423601154842827416136640203335, −9.299069256035598976488010760525, −8.653978763543362151115276256575, −8.609865643687584145814140494184, −7.71727870379861003800066083423, −7.50036789352519607150077351589, −7.35815629138189633660207703826, −6.69711758405768371451628293532, −6.51697029582375358696865609159, −6.00492176062609971528052945382, −5.33998168833763295369294349275, −5.25872329857534446272745279929, −4.19721173499108813684478401692, −3.96995804489005579109246465406, −3.54131370540973753219953631174, −3.02677733248705294704741263461, −2.91853978046178974766531513229, −1.88869475893590461747925331408, −1.40586227760554875656016455010, −0.965181331712455148186417555623,
0.965181331712455148186417555623, 1.40586227760554875656016455010, 1.88869475893590461747925331408, 2.91853978046178974766531513229, 3.02677733248705294704741263461, 3.54131370540973753219953631174, 3.96995804489005579109246465406, 4.19721173499108813684478401692, 5.25872329857534446272745279929, 5.33998168833763295369294349275, 6.00492176062609971528052945382, 6.51697029582375358696865609159, 6.69711758405768371451628293532, 7.35815629138189633660207703826, 7.50036789352519607150077351589, 7.71727870379861003800066083423, 8.609865643687584145814140494184, 8.653978763543362151115276256575, 9.299069256035598976488010760525, 9.423601154842827416136640203335
