L(s) = 1 | + 4-s + 8·5-s + 9-s − 6·11-s − 2·19-s + 8·20-s + 38·25-s + 4·29-s + 16·31-s + 36-s − 20·41-s − 6·44-s + 8·45-s + 11·49-s − 48·55-s − 64-s − 24·71-s − 2·76-s + 56·79-s − 2·89-s − 16·95-s − 6·99-s + 38·100-s − 20·101-s − 40·109-s + 4·116-s + 31·121-s + ⋯ |
L(s) = 1 | + 1/2·4-s + 3.57·5-s + 1/3·9-s − 1.80·11-s − 0.458·19-s + 1.78·20-s + 38/5·25-s + 0.742·29-s + 2.87·31-s + 1/6·36-s − 3.12·41-s − 0.904·44-s + 1.19·45-s + 11/7·49-s − 6.47·55-s − 1/8·64-s − 2.84·71-s − 0.229·76-s + 6.30·79-s − 0.211·89-s − 1.64·95-s − 0.603·99-s + 19/5·100-s − 1.99·101-s − 3.83·109-s + 0.371·116-s + 2.81·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.411925206\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.411925206\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
good | 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 - 7 T^{2} - 1320 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 58 T^{2} + 1515 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 45 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 98 T^{2} + 5115 T^{4} + 98 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 12 T + 73 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + T - 88 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 94 T^{2} - 573 T^{4} + 94 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.296866411723467936506455195064, −7.980149495280155477927860625146, −7.82830899626348540891067478516, −7.40237288842277226922411069453, −6.87589049076241351112091511031, −6.81366895084444165711807589151, −6.74976605661738953400008121158, −6.46468746572719070163562776483, −6.02870345892483446219302293580, −6.02036717452633514760973348306, −5.72370042494643021943419985644, −5.40405108149311667109437889156, −5.01761637383860217653087170267, −5.01244503294151166544353413361, −4.80783387313018461080310111111, −4.42960344148760180588775099541, −3.91103772676776058047965375344, −3.33920240179346243036148945605, −3.04412745765232313891655648181, −2.60553318104467224007780826749, −2.53687819667944540070053991858, −2.25732713056652101146486863066, −1.80932312091112183019915576553, −1.45814457036078424350368723308, −0.904731430141419157061923437734,
0.904731430141419157061923437734, 1.45814457036078424350368723308, 1.80932312091112183019915576553, 2.25732713056652101146486863066, 2.53687819667944540070053991858, 2.60553318104467224007780826749, 3.04412745765232313891655648181, 3.33920240179346243036148945605, 3.91103772676776058047965375344, 4.42960344148760180588775099541, 4.80783387313018461080310111111, 5.01244503294151166544353413361, 5.01761637383860217653087170267, 5.40405108149311667109437889156, 5.72370042494643021943419985644, 6.02036717452633514760973348306, 6.02870345892483446219302293580, 6.46468746572719070163562776483, 6.74976605661738953400008121158, 6.81366895084444165711807589151, 6.87589049076241351112091511031, 7.40237288842277226922411069453, 7.82830899626348540891067478516, 7.980149495280155477927860625146, 8.296866411723467936506455195064