| L(s) = 1 | + 1.41i·2-s + 1.73·3-s − 2.00·4-s − 2.72·5-s + 2.44i·6-s − 3.73·7-s − 2.82i·8-s + 2.99·9-s − 3.84i·10-s + 14.1i·11-s − 3.46·12-s + 1.27i·13-s − 5.28i·14-s − 4.71·15-s + 4.00·16-s − 24.7·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.500·4-s − 0.544·5-s + 0.408i·6-s − 0.533·7-s − 0.353i·8-s + 0.333·9-s − 0.384i·10-s + 1.28i·11-s − 0.288·12-s + 0.0981i·13-s − 0.377i·14-s − 0.314·15-s + 0.250·16-s − 1.45·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.197i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0619329 - 0.622432i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0619329 - 0.622432i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 - 1.73T \) |
| 59 | \( 1 + (-11.6 - 57.8i)T \) |
| good | 5 | \( 1 + 2.72T + 25T^{2} \) |
| 7 | \( 1 + 3.73T + 49T^{2} \) |
| 11 | \( 1 - 14.1iT - 121T^{2} \) |
| 13 | \( 1 - 1.27iT - 169T^{2} \) |
| 17 | \( 1 + 24.7T + 289T^{2} \) |
| 19 | \( 1 + 17.2T + 361T^{2} \) |
| 23 | \( 1 - 4.11iT - 529T^{2} \) |
| 29 | \( 1 + 30.6T + 841T^{2} \) |
| 31 | \( 1 + 11.8iT - 961T^{2} \) |
| 37 | \( 1 - 24.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 36.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 23.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 64.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 77.6T + 2.80e3T^{2} \) |
| 61 | \( 1 - 25.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 72.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 18.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + 99.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 34.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 48.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 51.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 68.4iT - 9.40e3T^{2} \) |
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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
−11.87902262591541398321140343267, −10.67082773431136756195601395095, −9.587984441241210814935732533427, −8.943668419281778725772533426967, −7.84713592427996632123290232159, −7.11415952282248722550211193358, −6.19212144169237999504666334720, −4.62949683543668687275881022488, −3.87929813434382368468209516090, −2.21624644600798704214933207609,
0.25070024108126706950804123181, 2.23772419258034906988491335416, 3.45513616942558421561987670351, 4.26279997018694107145840600623, 5.82164143514291403020638942507, 7.02222431033859033095757323870, 8.316804900029484765509018508272, 8.833945599356042094034176992860, 9.855571911059750330262418888785, 10.94882888883374576295252852950
