L(s) = 1 | + 2-s + (1.41 − 1.41i)3-s + 4-s + (0.707 + 2.12i)5-s + (1.41 − 1.41i)6-s + (1.29 − 1.29i)7-s + 8-s − 1.00i·9-s + (0.707 + 2.12i)10-s + 1.82i·11-s + (1.41 − 1.41i)12-s − 6.24·13-s + (1.29 − 1.29i)14-s + (4 + 1.99i)15-s + 16-s − 3.82i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.816 − 0.816i)3-s + 0.5·4-s + (0.316 + 0.948i)5-s + (0.577 − 0.577i)6-s + (0.488 − 0.488i)7-s + 0.353·8-s − 0.333i·9-s + (0.223 + 0.670i)10-s + 0.551i·11-s + (0.408 − 0.408i)12-s − 1.73·13-s + (0.345 − 0.345i)14-s + (1.03 + 0.516i)15-s + 0.250·16-s − 0.928i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.61201 - 0.414021i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.61201 - 0.414021i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + (-0.707 - 2.12i)T \) |
| 37 | \( 1 + (3.53 + 4.94i)T \) |
good | 3 | \( 1 + (-1.41 + 1.41i)T - 3iT^{2} \) |
| 7 | \( 1 + (-1.29 + 1.29i)T - 7iT^{2} \) |
| 11 | \( 1 - 1.82iT - 11T^{2} \) |
| 13 | \( 1 + 6.24T + 13T^{2} \) |
| 17 | \( 1 + 3.82iT - 17T^{2} \) |
| 19 | \( 1 + (-0.171 - 0.171i)T + 19iT^{2} \) |
| 23 | \( 1 + 5.41T + 23T^{2} \) |
| 29 | \( 1 + (-5.29 + 5.29i)T - 29iT^{2} \) |
| 31 | \( 1 + (-0.121 - 0.121i)T + 31iT^{2} \) |
| 41 | \( 1 + 7iT - 41T^{2} \) |
| 43 | \( 1 + 7T + 43T^{2} \) |
| 47 | \( 1 + (-0.242 + 0.242i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.12 + 2.12i)T + 53iT^{2} \) |
| 59 | \( 1 + (-6.82 - 6.82i)T + 59iT^{2} \) |
| 61 | \( 1 + (-10.7 - 10.7i)T + 61iT^{2} \) |
| 67 | \( 1 + (-7.24 - 7.24i)T + 67iT^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + (6 - 6i)T - 73iT^{2} \) |
| 79 | \( 1 + (-1.65 - 1.65i)T + 79iT^{2} \) |
| 83 | \( 1 + (-4.82 - 4.82i)T + 83iT^{2} \) |
| 89 | \( 1 + (4.58 - 4.58i)T - 89iT^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−11.64265160033865414175365389346, −10.38320691111141527980330636321, −9.747286884078717412066433768017, −8.197829083070458488490163786106, −7.18144812819960075616334839422, −7.07044190748736069756951874833, −5.47223128330950021218278473162, −4.27153199826017436133056020815, −2.74363814732258187064109732597, −2.08513470684197744533469345544,
2.04926673312470587617590228006, 3.34956284265200716544459094383, 4.58666013020708072058807321483, 5.18307520199737626130753350306, 6.43211338321903990418147799861, 8.048651734215428306899751421560, 8.596558859571141880330200442739, 9.686823563070493776798197407820, 10.28640424309537244783467377120, 11.75454918749987171530844110417