L(s) = 1 | + (−1 + 1.73i)2-s + (1.5 + 2.59i)3-s + (−1.99 − 3.46i)4-s + (3.70 − 6.42i)5-s − 6·6-s + 7.99·8-s + (−4.5 + 7.79i)9-s + (7.41 + 12.8i)10-s + (−5.24 − 9.08i)11-s + (6.00 − 10.3i)12-s − 2.78·13-s + 22.2·15-s + (−8 + 13.8i)16-s + (25.2 + 43.6i)17-s + (−9 − 15.5i)18-s + (62.5 − 108. i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.331 − 0.574i)5-s − 0.408·6-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.234 + 0.406i)10-s + (−0.143 − 0.248i)11-s + (0.144 − 0.249i)12-s − 0.0594·13-s + 0.382·15-s + (−0.125 + 0.216i)16-s + (0.359 + 0.623i)17-s + (−0.117 − 0.204i)18-s + (0.754 − 1.30i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.753888857\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.753888857\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - 1.73i)T \) |
| 3 | \( 1 + (-1.5 - 2.59i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-3.70 + 6.42i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (5.24 + 9.08i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 2.78T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-25.2 - 43.6i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-62.5 + 108. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-91.1 + 157. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 156.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-69.8 - 120. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-197. + 341. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 197.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 343.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (305. - 528. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-68.7 - 119. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-294. - 510. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-123. + 214. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-197. - 342. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 285.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (498. + 863. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-424. + 734. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 210.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (276. - 479. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 903.T + 9.12e5T^{2} \) |
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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
−11.11102832063902465056263176582, −10.31963604220193784219254894638, −9.226995306421794453178332391108, −8.750171119304606844383971522934, −7.67024916278719500666449591141, −6.50180842657187499744186473756, −5.31910102700749610128031135464, −4.48958461364012067884978525250, −2.82009392903557096189334896203, −0.912251677926634142598715966588,
1.14342501457630612979047188323, 2.51503878560418890592643173052, 3.52558962983131916515414114309, 5.18854720306154847801049812726, 6.53688901837956724891541769642, 7.54211901128197272610444229434, 8.376735817337864885481865251050, 9.686968054486674227963614560919, 10.10767887698802612357824042074, 11.41655601124922195955505199975