| L(s) = 1 | + (0.448 − 1.34i)2-s + (1.08 + 0.505i)3-s + (−1.59 − 1.20i)4-s + (1.58 + 2.26i)5-s + (1.16 − 1.22i)6-s + (2.37 + 4.11i)7-s + (−2.33 + 1.60i)8-s + (−1.00 − 1.20i)9-s + (3.75 − 1.11i)10-s + (1.48 − 0.398i)11-s + (−1.12 − 2.11i)12-s + (−1.57 + 0.735i)13-s + (6.57 − 1.33i)14-s + (0.574 + 3.25i)15-s + (1.10 + 3.84i)16-s + (0.0982 + 0.0824i)17-s + ⋯ |
| L(s) = 1 | + (0.317 − 0.948i)2-s + (0.625 + 0.291i)3-s + (−0.798 − 0.602i)4-s + (0.709 + 1.01i)5-s + (0.475 − 0.500i)6-s + (0.896 + 1.55i)7-s + (−0.824 + 0.566i)8-s + (−0.336 − 0.400i)9-s + (1.18 − 0.351i)10-s + (0.448 − 0.120i)11-s + (−0.324 − 0.609i)12-s + (−0.437 + 0.203i)13-s + (1.75 − 0.357i)14-s + (0.148 + 0.841i)15-s + (0.275 + 0.961i)16-s + (0.0238 + 0.0199i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.88924 - 0.267910i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.88924 - 0.267910i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.448 + 1.34i)T \) |
| 19 | \( 1 + (1.86 + 3.93i)T \) |
| good | 3 | \( 1 + (-1.08 - 0.505i)T + (1.92 + 2.29i)T^{2} \) |
| 5 | \( 1 + (-1.58 - 2.26i)T + (-1.71 + 4.69i)T^{2} \) |
| 7 | \( 1 + (-2.37 - 4.11i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.48 + 0.398i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (1.57 - 0.735i)T + (8.35 - 9.95i)T^{2} \) |
| 17 | \( 1 + (-0.0982 - 0.0824i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (1.22 + 6.95i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.176 + 2.01i)T + (-28.5 - 5.03i)T^{2} \) |
| 31 | \( 1 + (2.61 + 4.52i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.76 - 2.76i)T + 37iT^{2} \) |
| 41 | \( 1 + (8.45 - 3.07i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-8.40 + 5.88i)T + (14.7 - 40.4i)T^{2} \) |
| 47 | \( 1 + (-4.25 - 5.07i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-3.66 + 5.23i)T + (-18.1 - 49.8i)T^{2} \) |
| 59 | \( 1 + (-6.28 + 0.549i)T + (58.1 - 10.2i)T^{2} \) |
| 61 | \( 1 + (4.52 - 6.46i)T + (-20.8 - 57.3i)T^{2} \) |
| 67 | \( 1 + (15.1 + 1.32i)T + (65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (-10.9 - 1.93i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-4.02 - 11.0i)T + (-55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (5.25 - 1.91i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.472 - 0.126i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (2.36 + 0.860i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (7.39 - 8.81i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−11.66046952949397073110009929671, −10.85977402767712896546558008919, −9.800157654370111530701920474585, −9.026150537621922077620795241426, −8.378587431756340706291961941513, −6.46013776275245475112330076492, −5.58817063981833641116179757987, −4.29323033072772103865822006122, −2.70881781812384830774142543112, −2.29981289051386536551464603966,
1.53044930751753919001493469161, 3.71825260423319410282054935849, 4.81124557092949871218124501384, 5.66450175744012101638978568335, 7.19065387021205556513172847447, 7.80885117858468761710115992178, 8.660999455701040065201988323534, 9.562951403450173347893942262102, 10.70480695753701038740735071217, 12.10551027029825738748599608784
