| L(s) = 1 | − 5·9-s − 4·13-s + 6·17-s − 20·25-s + 10·29-s − 18·37-s + 6·41-s + 3·49-s − 32·53-s + 18·61-s − 16·73-s + 9·81-s + 2·89-s − 14·97-s + 30·101-s − 64·109-s + 22·113-s + 20·117-s + 11·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 30·153-s + 157-s + ⋯ |
| L(s) = 1 | − 5/3·9-s − 1.10·13-s + 1.45·17-s − 4·25-s + 1.85·29-s − 2.95·37-s + 0.937·41-s + 3/7·49-s − 4.39·53-s + 2.30·61-s − 1.87·73-s + 81-s + 0.211·89-s − 1.42·97-s + 2.98·101-s − 6.13·109-s + 2.06·113-s + 1.84·117-s + 121-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 − 2.42·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2152903485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2152903485\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 3 | $C_2^3$ | \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 7 | $C_2^3$ | \( 1 - 3 T^{2} - 40 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 - 27 T^{2} + 368 T^{4} - 27 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 35 T^{2} + 696 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 13 T^{2} - 1680 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 - 107 T^{2} + 7968 T^{4} - 107 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 9 T + 20 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 35 T^{2} - 3264 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^3$ | \( 1 - 43 T^{2} - 3192 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 114 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - T - 88 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.137168737027138719099948848284, −7.87273121172963451066923323974, −7.60955100485619283118037922508, −7.60017885510005422526492770268, −7.05266527011519654170626685934, −7.05029437252841768071911368005, −6.59656335498201544698006380011, −6.20630909665338915881201577156, −6.03063996788653119022733098805, −5.92359059338008495314646692299, −5.66840111213850985847655353908, −5.15945733075600577077676271597, −5.14897833310567816177935343407, −4.98119725872838407342100120594, −4.38431073096514722446644601749, −4.20983021190472638747178398866, −3.73067982323631693501141951886, −3.53363272033310291419500223737, −3.18986675445017056955313795116, −2.94841899390525123404647264971, −2.48349446992287228183830790898, −2.21698153858537655787529738817, −1.72082436000843088939663263624, −1.31218675849413734944107603764, −0.16880055800898607543769502140,
0.16880055800898607543769502140, 1.31218675849413734944107603764, 1.72082436000843088939663263624, 2.21698153858537655787529738817, 2.48349446992287228183830790898, 2.94841899390525123404647264971, 3.18986675445017056955313795116, 3.53363272033310291419500223737, 3.73067982323631693501141951886, 4.20983021190472638747178398866, 4.38431073096514722446644601749, 4.98119725872838407342100120594, 5.14897833310567816177935343407, 5.15945733075600577077676271597, 5.66840111213850985847655353908, 5.92359059338008495314646692299, 6.03063996788653119022733098805, 6.20630909665338915881201577156, 6.59656335498201544698006380011, 7.05029437252841768071911368005, 7.05266527011519654170626685934, 7.60017885510005422526492770268, 7.60955100485619283118037922508, 7.87273121172963451066923323974, 8.137168737027138719099948848284
