| L(s) = 1 | + 4-s − 2·5-s + 9-s − 2·20-s + 25-s + 36-s − 8·41-s − 2·45-s − 64-s + 4·89-s + 100-s − 4·101-s − 4·109-s + 2·121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 8·164-s + 167-s − 4·169-s + 173-s + ⋯ |
| L(s) = 1 | + 4-s − 2·5-s + 9-s − 2·20-s + 25-s + 36-s − 8·41-s − 2·45-s − 64-s + 4·89-s + 100-s − 4·101-s − 4·109-s + 2·121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 8·164-s + 167-s − 4·169-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1800408301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1800408301\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$ | \( ( 1 + T )^{8} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−6.50541707819806403067222036354, −6.23821146358369771826364324617, −6.23012269315381793809672239792, −5.85533585277973196116405765934, −5.46641493463402487236926179943, −5.21132738450774652616288641373, −5.15292917365827369586545621938, −4.91686797166519555421729354156, −4.90005316796233294813925979236, −4.54802536982248307831796914376, −4.22167165803769342033650478138, −3.99134073744599441017749522013, −3.83986521937622348563798421224, −3.80168843215814842436338145467, −3.32072661100767138967247509774, −3.31885216328517251685205956689, −3.02437970030273387139835011371, −2.95142353047164749026757425566, −2.39640719072856977584623391730, −2.08700406865543474991601139592, −1.84742648054615767791885384897, −1.78290867171680246231066947086, −1.23654153983932471407331482978, −1.21202480662362672384843919771, −0.16593468439315686064866122312,
0.16593468439315686064866122312, 1.21202480662362672384843919771, 1.23654153983932471407331482978, 1.78290867171680246231066947086, 1.84742648054615767791885384897, 2.08700406865543474991601139592, 2.39640719072856977584623391730, 2.95142353047164749026757425566, 3.02437970030273387139835011371, 3.31885216328517251685205956689, 3.32072661100767138967247509774, 3.80168843215814842436338145467, 3.83986521937622348563798421224, 3.99134073744599441017749522013, 4.22167165803769342033650478138, 4.54802536982248307831796914376, 4.90005316796233294813925979236, 4.91686797166519555421729354156, 5.15292917365827369586545621938, 5.21132738450774652616288641373, 5.46641493463402487236926179943, 5.85533585277973196116405765934, 6.23012269315381793809672239792, 6.23821146358369771826364324617, 6.50541707819806403067222036354
