| L(s) = 1 | + 25.6·2-s − 189.·3-s + 148.·4-s − 1.73e3·5-s − 4.86e3·6-s + 2.77e3·7-s − 9.34e3·8-s + 1.61e4·9-s − 4.46e4·10-s + 7.76e4·11-s − 2.80e4·12-s + 2.61e4·13-s + 7.12e4·14-s + 3.29e5·15-s − 3.16e5·16-s − 2.08e5·17-s + 4.14e5·18-s + 8.02e5·19-s − 2.57e5·20-s − 5.24e5·21-s + 1.99e6·22-s + 9.65e5·23-s + 1.76e6·24-s + 1.06e6·25-s + 6.71e5·26-s + 6.69e5·27-s + 4.10e5·28-s + ⋯ |
| L(s) = 1 | + 1.13·2-s − 1.34·3-s + 0.289·4-s − 1.24·5-s − 1.53·6-s + 0.436·7-s − 0.806·8-s + 0.820·9-s − 1.41·10-s + 1.59·11-s − 0.390·12-s + 0.253·13-s + 0.495·14-s + 1.67·15-s − 1.20·16-s − 0.606·17-s + 0.931·18-s + 1.41·19-s − 0.360·20-s − 0.588·21-s + 1.81·22-s + 0.719·23-s + 1.08·24-s + 0.546·25-s + 0.288·26-s + 0.242·27-s + 0.126·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.505297555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.505297555\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 - 3.41e6T \) |
| good | 2 | \( 1 - 25.6T + 512T^{2} \) |
| 3 | \( 1 + 189.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.73e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 2.77e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.76e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 2.61e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.08e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.02e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 9.65e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.78e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.47e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.14e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.01e7T + 3.27e14T^{2} \) |
| 47 | \( 1 - 4.43e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 7.35e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 4.64e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.71e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 5.65e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.28e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.92e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.58e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.23e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.51e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 8.11e8T + 7.60e17T^{2} \) |
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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
−13.97108121201139529113827329055, −12.47673423691477887788563111331, −11.65459384747513365183104880636, −11.24096525948291504369015772783, −9.039682496042384349148885046381, −7.14251851304890181321387174647, −5.86481431020994654782838736352, −4.64308785009378208959205102822, −3.67250026911332544519879333959, −0.76118027605914382296865303310,
0.76118027605914382296865303310, 3.67250026911332544519879333959, 4.64308785009378208959205102822, 5.86481431020994654782838736352, 7.14251851304890181321387174647, 9.039682496042384349148885046381, 11.24096525948291504369015772783, 11.65459384747513365183104880636, 12.47673423691477887788563111331, 13.97108121201139529113827329055
