| L(s) = 1 | − 16-s + 14·49-s + 16·67-s + 32·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | − 1/4·16-s + 2·49-s + 1.95·67-s + 3.60·79-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.477066772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.477066772\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 + T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 206 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - 734 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 1234 T^{4} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^3$ | \( 1 - 5582 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 71 | $C_2^3$ | \( 1 + 2914 T^{4} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.71595373065199350240138613809, −6.67187287926856091829365227287, −6.33530301830925455753161773921, −5.85220759204278267341394334634, −5.83159702892474670142972799150, −5.76379927290022681150116843764, −5.60678939072480557305574146205, −4.95324942122393747081156992276, −4.90611365990112501245130557397, −4.88865802796263248322511255460, −4.69083935676674260424930231941, −4.09724138510205194607742399311, −4.07729096095396247287052080441, −3.71174307494864839037167559787, −3.69595171713996581894541618111, −3.29524601271060746430909603846, −3.05769001791890848389834561012, −2.70742386067684719784832380015, −2.42209821186719376666119312660, −2.22011301085227501997125388417, −1.88212149471879291596206431913, −1.71436586487546910800778629431, −1.01324772106278416783974123480, −0.830088934259370455168746585630, −0.42818433567125402073333781455,
0.42818433567125402073333781455, 0.830088934259370455168746585630, 1.01324772106278416783974123480, 1.71436586487546910800778629431, 1.88212149471879291596206431913, 2.22011301085227501997125388417, 2.42209821186719376666119312660, 2.70742386067684719784832380015, 3.05769001791890848389834561012, 3.29524601271060746430909603846, 3.69595171713996581894541618111, 3.71174307494864839037167559787, 4.07729096095396247287052080441, 4.09724138510205194607742399311, 4.69083935676674260424930231941, 4.88865802796263248322511255460, 4.90611365990112501245130557397, 4.95324942122393747081156992276, 5.60678939072480557305574146205, 5.76379927290022681150116843764, 5.83159702892474670142972799150, 5.85220759204278267341394334634, 6.33530301830925455753161773921, 6.67187287926856091829365227287, 6.71595373065199350240138613809
