L(s) = 1 | + 39.9·2-s + 192.·3-s + 1.08e3·4-s − 2.27e3·5-s + 7.67e3·6-s + 1.05e4·7-s + 2.27e4·8-s + 1.72e4·9-s − 9.06e4·10-s + 2.61e4·11-s + 2.07e5·12-s + 1.30e5·13-s + 4.20e5·14-s − 4.36e5·15-s + 3.55e5·16-s + 6.88e5·18-s − 4.46e5·19-s − 2.45e6·20-s + 2.02e6·21-s + 1.04e6·22-s − 3.40e5·23-s + 4.37e6·24-s + 3.19e6·25-s + 5.21e6·26-s − 4.69e5·27-s + 1.13e7·28-s + 3.01e6·29-s + ⋯ |
L(s) = 1 | + 1.76·2-s + 1.36·3-s + 2.11·4-s − 1.62·5-s + 2.41·6-s + 1.65·7-s + 1.96·8-s + 0.875·9-s − 2.86·10-s + 0.539·11-s + 2.89·12-s + 1.26·13-s + 2.92·14-s − 2.22·15-s + 1.35·16-s + 1.54·18-s − 0.785·19-s − 3.43·20-s + 2.26·21-s + 0.951·22-s − 0.253·23-s + 2.69·24-s + 1.63·25-s + 2.23·26-s − 0.169·27-s + 3.50·28-s + 0.791·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(12.06316506\) |
\(L(\frac12)\) |
\(\approx\) |
\(12.06316506\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
good | 2 | \( 1 - 39.9T + 512T^{2} \) |
| 3 | \( 1 - 192.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.27e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.05e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.61e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.30e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 4.46e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 3.40e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.01e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.67e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.57e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.68e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.62e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.82e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.06e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 5.56e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.23e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.03e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.91e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.25e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 6.19e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.60e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.09e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 5.75e8T + 7.60e17T^{2} \) |
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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
−10.88149642429056564508482363695, −8.775552729263615042682698755218, −8.112197056101894069824011636303, −7.48459525266305518516657441338, −6.19817935178976865701075353763, −4.65705328662507455652189605401, −4.14245588201294160355544759957, −3.45976837045708516718417700889, −2.38941914047985883364551134955, −1.24319309737845115806007063936,
1.24319309737845115806007063936, 2.38941914047985883364551134955, 3.45976837045708516718417700889, 4.14245588201294160355544759957, 4.65705328662507455652189605401, 6.19817935178976865701075353763, 7.48459525266305518516657441338, 8.112197056101894069824011636303, 8.775552729263615042682698755218, 10.88149642429056564508482363695