| L(s) = 1 | + (1.33e3 + 2.32e3i)5-s + (−2.80e4 − 3.45e4i)7-s + (2.39e5 − 4.14e5i)11-s − 1.17e6·13-s + (2.40e6 − 4.17e6i)17-s + (−4.55e6 − 7.88e6i)19-s + (4.21e6 + 7.30e6i)23-s + (2.08e7 − 3.60e7i)25-s − 3.65e7·29-s + (−1.20e7 + 2.09e7i)31-s + (4.25e7 − 1.11e8i)35-s + (−1.66e8 − 2.88e8i)37-s − 5.86e7·41-s + 8.57e8·43-s + (−4.75e8 − 8.24e8i)47-s + ⋯ |
| L(s) = 1 | + (0.191 + 0.332i)5-s + (−0.630 − 0.776i)7-s + (0.448 − 0.776i)11-s − 0.874·13-s + (0.411 − 0.712i)17-s + (−0.421 − 0.730i)19-s + (0.136 + 0.236i)23-s + (0.426 − 0.738i)25-s − 0.331·29-s + (−0.0758 + 0.131i)31-s + (0.136 − 0.358i)35-s + (−0.395 − 0.684i)37-s − 0.0791·41-s + 0.889·43-s + (−0.302 − 0.524i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 - 0.415i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.4329645579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4329645579\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.80e4 + 3.45e4i)T \) |
| good | 5 | \( 1 + (-1.33e3 - 2.32e3i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-2.39e5 + 4.14e5i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 + 1.17e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-2.40e6 + 4.17e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (4.55e6 + 7.88e6i)T + (-5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-4.21e6 - 7.30e6i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + 3.65e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + (1.20e7 - 2.09e7i)T + (-1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 + (1.66e8 + 2.88e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 + 5.86e7T + 5.50e17T^{2} \) |
| 43 | \( 1 - 8.57e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (4.75e8 + 8.24e8i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-9.09e8 + 1.57e9i)T + (-4.63e18 - 8.02e18i)T^{2} \) |
| 59 | \( 1 + (8.43e8 - 1.46e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-3.82e9 - 6.63e9i)T + (-2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (2.92e9 - 5.06e9i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 + 1.03e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (-4.61e9 + 7.99e9i)T + (-1.56e20 - 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-9.94e9 - 1.72e10i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 + 4.78e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-9.70e9 - 1.68e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 - 8.32e10T + 7.15e21T^{2} \) |
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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
−9.647580598823320995417402353088, −8.720057008033691222278997783433, −7.38506193278918106518150227901, −6.76798576690229806594992863905, −5.67368699637365890148083200547, −4.45102263979308960806077684374, −3.35816421007057691968930451356, −2.44164856963942049923350134100, −0.922647806466379570406600765826, −0.088875144554082774384085114930,
1.40471053283801573920350541802, 2.35764170945433944545472741870, 3.55244365868081267767661338126, 4.74974655114531175137983415028, 5.74374352175596945017690177989, 6.68283221712383260655622436230, 7.77819537934536556334004553001, 8.912161268904668061308123416510, 9.618617598882890280925286998788, 10.44638287618238496363799039187
