| L(s) = 1 | + 3-s + 3·4-s + 9-s + 3·12-s + 2·13-s + 5·16-s − 25-s + 27-s + 3·36-s + 2·39-s + 8·43-s + 5·48-s − 14·49-s + 6·52-s + 4·61-s + 3·64-s − 75-s + 81-s − 3·100-s − 32·103-s + 3·108-s + 2·117-s + 6·121-s + 127-s + 8·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 3/2·4-s + 1/3·9-s + 0.866·12-s + 0.554·13-s + 5/4·16-s − 1/5·25-s + 0.192·27-s + 1/2·36-s + 0.320·39-s + 1.21·43-s + 0.721·48-s − 2·49-s + 0.832·52-s + 0.512·61-s + 3/8·64-s − 0.115·75-s + 1/9·81-s − 0.299·100-s − 3.15·103-s + 0.288·108-s + 0.184·117-s + 6/11·121-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114075 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114075 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.863990301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.863990301\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.564474286580122427495577154992, −8.980294450608097651050448300689, −8.351409413861350040873257357178, −7.991681751842688113770380535684, −7.51628611970644146916073374604, −6.92839021493812478601438926742, −6.62450271112658542252207149269, −6.01789423967056470753162236290, −5.56849171211806971312774945579, −4.74451546343375410691487409945, −4.02431384528355904678620743579, −3.35431190377298422152925553138, −2.77611484237904402338173740434, −2.10875221203602717470646927840, −1.34931823293579996397719805792,
1.34931823293579996397719805792, 2.10875221203602717470646927840, 2.77611484237904402338173740434, 3.35431190377298422152925553138, 4.02431384528355904678620743579, 4.74451546343375410691487409945, 5.56849171211806971312774945579, 6.01789423967056470753162236290, 6.62450271112658542252207149269, 6.92839021493812478601438926742, 7.51628611970644146916073374604, 7.991681751842688113770380535684, 8.351409413861350040873257357178, 8.980294450608097651050448300689, 9.564474286580122427495577154992
