| L(s) = 1 | + (−0.490 − 1.32i)2-s + (−2.34 + 0.264i)3-s + (−1.51 + 1.30i)4-s + (−0.588 + 1.22i)5-s + (1.50 + 2.98i)6-s + (2.56 + 2.04i)7-s + (2.47 + 1.37i)8-s + (2.52 − 0.575i)9-s + (1.91 + 0.181i)10-s + (−5.07 + 3.19i)11-s + (3.22 − 3.45i)12-s + (−1.29 − 0.296i)13-s + (1.45 − 4.39i)14-s + (1.05 − 3.02i)15-s + (0.615 − 3.95i)16-s + (−3.79 + 3.79i)17-s + ⋯ |
| L(s) = 1 | + (−0.346 − 0.937i)2-s + (−1.35 + 0.152i)3-s + (−0.759 + 0.650i)4-s + (−0.263 + 0.546i)5-s + (0.613 + 1.21i)6-s + (0.968 + 0.772i)7-s + (0.873 + 0.486i)8-s + (0.840 − 0.191i)9-s + (0.604 + 0.0573i)10-s + (−1.53 + 0.962i)11-s + (0.930 − 0.998i)12-s + (−0.359 − 0.0821i)13-s + (0.388 − 1.17i)14-s + (0.273 − 0.781i)15-s + (0.153 − 0.988i)16-s + (−0.919 + 0.919i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.341 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.312544 + 0.218930i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.312544 + 0.218930i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.490 + 1.32i)T \) |
| 29 | \( 1 + (-2.93 + 4.51i)T \) |
| good | 3 | \( 1 + (2.34 - 0.264i)T + (2.92 - 0.667i)T^{2} \) |
| 5 | \( 1 + (0.588 - 1.22i)T + (-3.11 - 3.90i)T^{2} \) |
| 7 | \( 1 + (-2.56 - 2.04i)T + (1.55 + 6.82i)T^{2} \) |
| 11 | \( 1 + (5.07 - 3.19i)T + (4.77 - 9.91i)T^{2} \) |
| 13 | \( 1 + (1.29 + 0.296i)T + (11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (3.79 - 3.79i)T - 17iT^{2} \) |
| 19 | \( 1 + (-0.151 + 1.34i)T + (-18.5 - 4.22i)T^{2} \) |
| 23 | \( 1 + (-1.29 - 2.68i)T + (-14.3 + 17.9i)T^{2} \) |
| 31 | \( 1 + (1.44 + 4.14i)T + (-24.2 + 19.3i)T^{2} \) |
| 37 | \( 1 + (3.02 - 4.81i)T + (-16.0 - 33.3i)T^{2} \) |
| 41 | \( 1 + (4.55 + 4.55i)T + 41iT^{2} \) |
| 43 | \( 1 + (-6.65 - 2.33i)T + (33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (0.260 + 0.414i)T + (-20.3 + 42.3i)T^{2} \) |
| 53 | \( 1 + (-5.75 - 2.77i)T + (33.0 + 41.4i)T^{2} \) |
| 59 | \( 1 + 6.26iT - 59T^{2} \) |
| 61 | \( 1 + (-1.38 - 12.2i)T + (-59.4 + 13.5i)T^{2} \) |
| 67 | \( 1 + (-0.856 - 3.75i)T + (-60.3 + 29.0i)T^{2} \) |
| 71 | \( 1 + (2.73 - 11.9i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-7.19 - 2.51i)T + (57.0 + 45.5i)T^{2} \) |
| 79 | \( 1 + (-2.90 + 4.62i)T + (-34.2 - 71.1i)T^{2} \) |
| 83 | \( 1 + (2.50 - 1.99i)T + (18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-12.1 + 4.23i)T + (69.5 - 55.4i)T^{2} \) |
| 97 | \( 1 + (-0.141 + 1.25i)T + (-94.5 - 21.5i)T^{2} \) |
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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
−13.27529317641222227306832421421, −12.33539607687439119492156043921, −11.48604338972315137737783879493, −10.84057274976576526471497506241, −10.04491617764103710375010495664, −8.476487547999900586495623286559, −7.29787523402278920438672632460, −5.46175720496265322153308624222, −4.58025810736369975644233136203, −2.35301355645904820615558320807,
0.54925817958332078324495285454, 4.77114294217893018285801994487, 5.24643588503547898886961615030, 6.68069057404575526009919823164, 7.76449064290503183991704218100, 8.754909729854625994031117013443, 10.56030603884577047683239302627, 10.93731770910138042664429062204, 12.33510696412655165691431732871, 13.45005779427224207853718031276
