| L(s) = 1 | + 32·2-s + 1.02e3·4-s − 5.76e3·5-s + 7.24e4·7-s + 3.27e4·8-s − 1.84e5·10-s + 4.08e5·11-s + 1.36e6·13-s + 2.31e6·14-s + 1.04e6·16-s − 5.42e6·17-s + 1.51e7·19-s − 5.90e6·20-s + 1.30e7·22-s + 5.21e7·23-s − 1.55e7·25-s + 4.37e7·26-s + 7.42e7·28-s − 1.18e8·29-s − 5.76e7·31-s + 3.35e7·32-s − 1.73e8·34-s − 4.17e8·35-s − 3.75e8·37-s + 4.85e8·38-s − 1.88e8·40-s − 8.56e8·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.825·5-s + 1.62·7-s + 0.353·8-s − 0.583·10-s + 0.765·11-s + 1.02·13-s + 1.15·14-s + 1/4·16-s − 0.926·17-s + 1.40·19-s − 0.412·20-s + 0.541·22-s + 1.69·23-s − 0.319·25-s + 0.722·26-s + 0.814·28-s − 1.07·29-s − 0.361·31-s + 0.176·32-s − 0.655·34-s − 1.34·35-s − 0.891·37-s + 0.993·38-s − 0.291·40-s − 1.15·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(3.026556785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.026556785\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{5} T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5766 T + p^{11} T^{2} \) |
| 7 | \( 1 - 10352 p T + p^{11} T^{2} \) |
| 11 | \( 1 - 408948 T + p^{11} T^{2} \) |
| 13 | \( 1 - 1367558 T + p^{11} T^{2} \) |
| 17 | \( 1 + 5422914 T + p^{11} T^{2} \) |
| 19 | \( 1 - 15166100 T + p^{11} T^{2} \) |
| 23 | \( 1 - 52194072 T + p^{11} T^{2} \) |
| 29 | \( 1 + 118581150 T + p^{11} T^{2} \) |
| 31 | \( 1 + 57652408 T + p^{11} T^{2} \) |
| 37 | \( 1 + 375985186 T + p^{11} T^{2} \) |
| 41 | \( 1 + 856316202 T + p^{11} T^{2} \) |
| 43 | \( 1 + 1245189172 T + p^{11} T^{2} \) |
| 47 | \( 1 - 1306762656 T + p^{11} T^{2} \) |
| 53 | \( 1 + 409556358 T + p^{11} T^{2} \) |
| 59 | \( 1 - 48862140 p T + p^{11} T^{2} \) |
| 61 | \( 1 - 5731767302 T + p^{11} T^{2} \) |
| 67 | \( 1 - 3893272244 T + p^{11} T^{2} \) |
| 71 | \( 1 - 9075890088 T + p^{11} T^{2} \) |
| 73 | \( 1 + 15571822822 T + p^{11} T^{2} \) |
| 79 | \( 1 + 30196762600 T + p^{11} T^{2} \) |
| 83 | \( 1 + 23135252628 T + p^{11} T^{2} \) |
| 89 | \( 1 - 25614819990 T + p^{11} T^{2} \) |
| 97 | \( 1 + 61937553406 T + p^{11} T^{2} \) |
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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
−15.66637610295019069721620381724, −14.71362713078052218175597707082, −13.49549699139115051091090101378, −11.68199947895336558783550157110, −11.15830835695510320294191419949, −8.626046372316532130994219306644, −7.19456034824773060150233749016, −5.13423970869104348744078534053, −3.74186047611681365877524602609, −1.43226230309828075489904425658,
1.43226230309828075489904425658, 3.74186047611681365877524602609, 5.13423970869104348744078534053, 7.19456034824773060150233749016, 8.626046372316532130994219306644, 11.15830835695510320294191419949, 11.68199947895336558783550157110, 13.49549699139115051091090101378, 14.71362713078052218175597707082, 15.66637610295019069721620381724
