| L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s + 3i·7-s − i·8-s − 9-s − 3·11-s + i·12-s + i·13-s − 3·14-s + 16-s − 2i·17-s − i·18-s + 3·21-s − 3i·22-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s + 1.13i·7-s − 0.353i·8-s − 0.333·9-s − 0.904·11-s + 0.288i·12-s + 0.277i·13-s − 0.801·14-s + 0.250·16-s − 0.485i·17-s − 0.235i·18-s + 0.654·21-s − 0.639i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.112466618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.112466618\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 37 | \( 1 - 3iT - 37T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 + 7iT - 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 7T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 9iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.151355040010086038132006048414, −7.55702145021278217422385953648, −6.81281298641150295904781099619, −6.09173466326245110948278167203, −5.42204040940358163004452734879, −4.89721879316322916544077808694, −3.72878244180412921300063031332, −2.66174572948670480986817453271, −1.97313796200454235915259665725, −0.38749659757673643003144463942,
0.852049747919853985150696849212, 2.06560487194614340859647218523, 3.07884597862881339300639645792, 3.85595633741555357093632899715, 4.36759882774457445660590204323, 5.35998835911126706062063377877, 5.87919262427791740700407167773, 7.15585865794075265081974401189, 7.70324868955234278426824855259, 8.377604799309773937457146028845
