L(s) = 1 | + 4.64i·2-s + 5.19·3-s − 5.59·4-s − 0.691·5-s + 24.1i·6-s − 76.1·7-s + 48.3i·8-s + 27·9-s − 3.21i·10-s + 74.8i·11-s − 29.0·12-s − 105. i·13-s − 353. i·14-s − 3.59·15-s − 314.·16-s − 141.·17-s + ⋯ |
L(s) = 1 | + 1.16i·2-s + 0.577·3-s − 0.349·4-s − 0.0276·5-s + 0.670i·6-s − 1.55·7-s + 0.755i·8-s + 0.333·9-s − 0.0321i·10-s + 0.618i·11-s − 0.201·12-s − 0.623i·13-s − 1.80i·14-s − 0.0159·15-s − 1.22·16-s − 0.491·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 + 0.742i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.7679528127\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7679528127\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 5.19T \) |
| 59 | \( 1 + (-2.33e3 + 2.58e3i)T \) |
good | 2 | \( 1 - 4.64iT - 16T^{2} \) |
| 5 | \( 1 + 0.691T + 625T^{2} \) |
| 7 | \( 1 + 76.1T + 2.40e3T^{2} \) |
| 11 | \( 1 - 74.8iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 105. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 141.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 170.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 226. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 677.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 114. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 488. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.82e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 527. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 1.90e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 4.68e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 6.41e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 1.77e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 170.T + 2.54e7T^{2} \) |
| 73 | \( 1 - 8.99e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 204.T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.39e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 3.00e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 9.22e3iT - 8.85e7T^{2} \) |
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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
−13.04238678787172880537717960291, −11.81269519333545519381097418012, −10.31889071465336622815524537981, −9.426584461979691733529637905153, −8.395120282249582070326599153875, −7.30512703646939716941723646928, −6.56390014537255418406585811798, −5.47780355516807496918609341358, −3.82142113625191082171316353814, −2.39122469662189332705227417569,
0.23803534299371464064052544405, 2.06216475363161757530614843390, 3.20530957504131990895225989747, 4.05682261182249838764560670611, 6.17474702682957273838224477314, 7.10883268240320283527190018391, 8.754947325997453833554240271648, 9.561947976636768320316145881956, 10.35060697366268595248184207690, 11.39086990378305255807987981529