| L(s) = 1 | + 6·3-s + 27·9-s + 20·11-s + 46·25-s + 108·27-s + 120·33-s − 2·49-s − 20·59-s − 100·73-s + 276·75-s + 405·81-s − 268·83-s − 380·97-s + 540·99-s + 172·107-s + 58·121-s + 127-s + 131-s + 137-s + 139-s − 12·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 338·169-s + ⋯ |
| L(s) = 1 | + 2·3-s + 3·9-s + 1.81·11-s + 1.83·25-s + 4·27-s + 3.63·33-s − 0.0408·49-s − 0.338·59-s − 1.36·73-s + 3.67·75-s + 5·81-s − 3.22·83-s − 3.91·97-s + 5.45·99-s + 1.60·107-s + 0.479·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 0.0816·147-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(6.450491057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.450491057\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 46 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 818 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 478 T^{2} + p^{4} T^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 3218 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 9118 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 134 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 190 T + p^{2} T^{2} )^{2} \) |
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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
−11.38735929574175280802255819663, −10.75292057700498591776000313541, −10.28257457368808273811594955813, −9.796777641915577562505696301354, −9.417570731552329298035594818010, −8.852233846252248000595043681143, −8.803810337409871934530914887229, −8.279699916020147107004656900770, −7.68659317271322669096255486016, −7.13652407948437661022276231243, −6.78405026053651202798906805311, −6.42483085051945135586144009216, −5.49053732849243540411472460185, −4.65763438272161243257722063711, −4.18567902089195589140381445356, −3.80477872015752117373927206569, −3.00628838253993141732650943564, −2.69725361932753444631051343939, −1.59472073085204128542764108900, −1.21647475445652756177773255622,
1.21647475445652756177773255622, 1.59472073085204128542764108900, 2.69725361932753444631051343939, 3.00628838253993141732650943564, 3.80477872015752117373927206569, 4.18567902089195589140381445356, 4.65763438272161243257722063711, 5.49053732849243540411472460185, 6.42483085051945135586144009216, 6.78405026053651202798906805311, 7.13652407948437661022276231243, 7.68659317271322669096255486016, 8.279699916020147107004656900770, 8.803810337409871934530914887229, 8.852233846252248000595043681143, 9.417570731552329298035594818010, 9.796777641915577562505696301354, 10.28257457368808273811594955813, 10.75292057700498591776000313541, 11.38735929574175280802255819663
