| L(s) = 1 | + 2·3-s + 4-s + 9-s + 2·12-s + 10·17-s + 12·23-s + 18·25-s − 2·27-s + 18·29-s + 36-s + 20·43-s − 10·49-s + 20·51-s − 4·53-s + 22·61-s − 64-s + 10·68-s + 24·69-s + 36·75-s − 16·79-s − 4·81-s + 36·87-s + 12·92-s + 18·100-s − 10·101-s − 40·103-s + 36·107-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 1/3·9-s + 0.577·12-s + 2.42·17-s + 2.50·23-s + 18/5·25-s − 0.384·27-s + 3.34·29-s + 1/6·36-s + 3.04·43-s − 1.42·49-s + 2.80·51-s − 0.549·53-s + 2.81·61-s − 1/8·64-s + 1.21·68-s + 2.88·69-s + 4.15·75-s − 1.80·79-s − 4/9·81-s + 3.85·87-s + 1.25·92-s + 9/5·100-s − 0.995·101-s − 3.94·103-s + 3.48·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.94212357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.94212357\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 9 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^3$ | \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 18 T^{2} + 203 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 + 34 T^{2} + 795 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 73 T^{2} + p^{2} T^{4} )( 1 + 26 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2^3$ | \( 1 + 57 T^{2} + 1568 T^{4} + 57 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 + 54 T^{2} - 565 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 130 T^{2} + 12411 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^3$ | \( 1 - 54 T^{2} - 2125 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 23 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 174 T^{2} + 22355 T^{4} + 174 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 190 T^{2} + 26691 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.32299905648529210536018500907, −6.89814275167028823054322365165, −6.60001802584233314163483371562, −6.55120266792438192233634043704, −6.51915324578637967625520898980, −6.04628845460694285616243849823, −5.66003858693035443548859144301, −5.57628344967917593115606222469, −5.25916793241391636916524349854, −4.95282952159382361517300713027, −4.90091955475211292105965876150, −4.56605556387738202044227674248, −4.50207464573377221875339068431, −3.88870813272920274318934168802, −3.66492497596857318336334144592, −3.51264861345915470364509317338, −3.04908599220719174876132770834, −2.91516793863115381196224805964, −2.64853706010199820781830749965, −2.60120632224401171121048076864, −2.48794050047420119202186078314, −1.50465139085063297129044890445, −1.23580470480409717011374851339, −1.00442739907000154704838934747, −0.861688325319878772331787383242,
0.861688325319878772331787383242, 1.00442739907000154704838934747, 1.23580470480409717011374851339, 1.50465139085063297129044890445, 2.48794050047420119202186078314, 2.60120632224401171121048076864, 2.64853706010199820781830749965, 2.91516793863115381196224805964, 3.04908599220719174876132770834, 3.51264861345915470364509317338, 3.66492497596857318336334144592, 3.88870813272920274318934168802, 4.50207464573377221875339068431, 4.56605556387738202044227674248, 4.90091955475211292105965876150, 4.95282952159382361517300713027, 5.25916793241391636916524349854, 5.57628344967917593115606222469, 5.66003858693035443548859144301, 6.04628845460694285616243849823, 6.51915324578637967625520898980, 6.55120266792438192233634043704, 6.60001802584233314163483371562, 6.89814275167028823054322365165, 7.32299905648529210536018500907
