| L(s) = 1 | + 5.45·2-s + 21.7·4-s + 5·5-s − 11.8·7-s + 75.3·8-s + 27.2·10-s − 56.2·11-s + 34.5·13-s − 64.4·14-s + 236.·16-s + 39.2·17-s − 146.·19-s + 108.·20-s − 306.·22-s − 23.5·23-s + 25·25-s + 188.·26-s − 257.·28-s + 161.·29-s − 29.5·31-s + 689.·32-s + 214.·34-s − 59.0·35-s − 217.·37-s − 800.·38-s + 376.·40-s + 142.·41-s + ⋯ |
| L(s) = 1 | + 1.92·2-s + 2.72·4-s + 0.447·5-s − 0.637·7-s + 3.32·8-s + 0.863·10-s − 1.54·11-s + 0.738·13-s − 1.23·14-s + 3.69·16-s + 0.560·17-s − 1.76·19-s + 1.21·20-s − 2.97·22-s − 0.213·23-s + 0.200·25-s + 1.42·26-s − 1.73·28-s + 1.03·29-s − 0.171·31-s + 3.81·32-s + 1.08·34-s − 0.285·35-s − 0.967·37-s − 3.41·38-s + 1.48·40-s + 0.541·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.926119094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.926119094\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 2 | \( 1 - 5.45T + 8T^{2} \) |
| 7 | \( 1 + 11.8T + 343T^{2} \) |
| 11 | \( 1 + 56.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 39.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 146.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 23.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 161.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 29.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 217.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 142.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 468.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 394.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 134.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 131.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 259.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 445.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 560.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 88.6T + 3.89e5T^{2} \) |
| 79 | \( 1 - 450.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 284.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 625.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 193.T + 9.12e5T^{2} \) |
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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
−12.97773233537716655234137043228, −12.24492162338329730772515500603, −10.86501632263758945479301866937, −10.25061735782117591736534269600, −8.171529742128186629101289393691, −6.74403561770058056568235677406, −5.90100708404861763999410979545, −4.85833407247916650396246333140, −3.46000291396205748825899277840, −2.26401802971586434376665794269,
2.26401802971586434376665794269, 3.46000291396205748825899277840, 4.85833407247916650396246333140, 5.90100708404861763999410979545, 6.74403561770058056568235677406, 8.171529742128186629101289393691, 10.25061735782117591736534269600, 10.86501632263758945479301866937, 12.24492162338329730772515500603, 12.97773233537716655234137043228
