| L(s) = 1 | − 156·3-s − 870·5-s + 952·7-s + 4.65e3·9-s − 5.61e4·11-s − 1.78e5·13-s + 1.35e5·15-s − 2.47e5·17-s + 3.15e5·19-s − 1.48e5·21-s − 2.04e5·23-s − 1.19e6·25-s + 2.34e6·27-s + 3.84e6·29-s + 1.30e6·31-s + 8.75e6·33-s − 8.28e5·35-s − 4.30e6·37-s + 2.77e7·39-s + 1.51e6·41-s + 3.36e7·43-s − 4.04e6·45-s + 1.05e7·47-s − 3.94e7·49-s + 3.86e7·51-s − 1.66e7·53-s + 4.88e7·55-s + ⋯ |
| L(s) = 1 | − 1.11·3-s − 0.622·5-s + 0.149·7-s + 0.236·9-s − 1.15·11-s − 1.72·13-s + 0.692·15-s − 0.719·17-s + 0.555·19-s − 0.166·21-s − 0.152·23-s − 0.612·25-s + 0.849·27-s + 1.00·29-s + 0.254·31-s + 1.28·33-s − 0.0932·35-s − 0.377·37-s + 1.92·39-s + 0.0835·41-s + 1.50·43-s − 0.147·45-s + 0.316·47-s − 0.977·49-s + 0.799·51-s − 0.289·53-s + 0.719·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.4685097937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4685097937\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 52 p T + p^{9} T^{2} \) |
| 5 | \( 1 + 174 p T + p^{9} T^{2} \) |
| 7 | \( 1 - 136 p T + p^{9} T^{2} \) |
| 11 | \( 1 + 56148 T + p^{9} T^{2} \) |
| 13 | \( 1 + 178094 T + p^{9} T^{2} \) |
| 17 | \( 1 + 247662 T + p^{9} T^{2} \) |
| 19 | \( 1 - 315380 T + p^{9} T^{2} \) |
| 23 | \( 1 + 204504 T + p^{9} T^{2} \) |
| 29 | \( 1 - 3840450 T + p^{9} T^{2} \) |
| 31 | \( 1 - 1309408 T + p^{9} T^{2} \) |
| 37 | \( 1 + 4307078 T + p^{9} T^{2} \) |
| 41 | \( 1 - 1512042 T + p^{9} T^{2} \) |
| 43 | \( 1 - 33670604 T + p^{9} T^{2} \) |
| 47 | \( 1 - 10581072 T + p^{9} T^{2} \) |
| 53 | \( 1 + 16616214 T + p^{9} T^{2} \) |
| 59 | \( 1 - 112235100 T + p^{9} T^{2} \) |
| 61 | \( 1 - 33197218 T + p^{9} T^{2} \) |
| 67 | \( 1 + 121372252 T + p^{9} T^{2} \) |
| 71 | \( 1 - 387172728 T + p^{9} T^{2} \) |
| 73 | \( 1 - 255240074 T + p^{9} T^{2} \) |
| 79 | \( 1 + 492101840 T + p^{9} T^{2} \) |
| 83 | \( 1 + 457420236 T + p^{9} T^{2} \) |
| 89 | \( 1 + 31809510 T + p^{9} T^{2} \) |
| 97 | \( 1 + 673532062 T + p^{9} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.64762925213974977196210315001, −11.86824234797208726183760268507, −10.92744646930088398347919999366, −9.846424508180237007999226868379, −8.105366524969710372053716914131, −7.01365576988175241137757914166, −5.50167552524074174689615383115, −4.57903610012729518589381775981, −2.58444646268313161836659331592, −0.42069829567333748762971291837,
0.42069829567333748762971291837, 2.58444646268313161836659331592, 4.57903610012729518589381775981, 5.50167552524074174689615383115, 7.01365576988175241137757914166, 8.105366524969710372053716914131, 9.846424508180237007999226868379, 10.92744646930088398347919999366, 11.86824234797208726183760268507, 12.64762925213974977196210315001
