L(s) = 1 | + 3.15·2-s + 1.96·4-s − 1.36·5-s + 13.0·7-s − 19.0·8-s − 4.30·10-s − 64.7·11-s − 19.2·13-s + 41.0·14-s − 75.8·16-s + 54.1·17-s − 69.0·19-s − 2.67·20-s − 204.·22-s − 29.6·23-s − 123.·25-s − 60.9·26-s + 25.5·28-s − 13.1·29-s + 185.·31-s − 87.0·32-s + 170.·34-s − 17.7·35-s − 369.·37-s − 218.·38-s + 25.9·40-s + 294.·41-s + ⋯ |
L(s) = 1 | + 1.11·2-s + 0.245·4-s − 0.121·5-s + 0.702·7-s − 0.842·8-s − 0.136·10-s − 1.77·11-s − 0.411·13-s + 0.784·14-s − 1.18·16-s + 0.772·17-s − 0.833·19-s − 0.0299·20-s − 1.98·22-s − 0.268·23-s − 0.985·25-s − 0.459·26-s + 0.172·28-s − 0.0840·29-s + 1.07·31-s − 0.480·32-s + 0.861·34-s − 0.0857·35-s − 1.64·37-s − 0.930·38-s + 0.102·40-s + 1.12·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 43 | \( 1 + 43T \) |
good | 2 | \( 1 - 3.15T + 8T^{2} \) |
| 5 | \( 1 + 1.36T + 125T^{2} \) |
| 7 | \( 1 - 13.0T + 343T^{2} \) |
| 11 | \( 1 + 64.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 19.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 54.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 69.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 29.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 13.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 185.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 369.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 294.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 367.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 708.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 116.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 218.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 133.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 926.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 455.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 620.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.31e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 509.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 965.T + 9.12e5T^{2} \) |
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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
−10.60512229301919488667779692919, −9.685776828962370226345696634049, −8.320449673406720848490907813222, −7.71988456766223119369219631848, −6.28317489401454061446700187726, −5.24830676940554805169476690130, −4.68542259197288803088603090618, −3.39783154111978565925044862582, −2.22891666055076259276258524745, 0,
2.22891666055076259276258524745, 3.39783154111978565925044862582, 4.68542259197288803088603090618, 5.24830676940554805169476690130, 6.28317489401454061446700187726, 7.71988456766223119369219631848, 8.320449673406720848490907813222, 9.685776828962370226345696634049, 10.60512229301919488667779692919