| L(s) = 1 | − i·2-s − 4-s − 2.44·5-s + i·8-s + 2.44i·10-s − 4.24i·11-s + 4.18i·13-s + 16-s − 2.44·17-s + 4.89i·19-s + 2.44·20-s − 4.24·22-s − 6i·23-s + 0.999·25-s + 4.18·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 1.09·5-s + 0.353i·8-s + 0.774i·10-s − 1.27i·11-s + 1.15i·13-s + 0.250·16-s − 0.594·17-s + 1.12i·19-s + 0.547·20-s − 0.904·22-s − 1.25i·23-s + 0.199·25-s + 0.820·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.073437887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.073437887\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2.44T + 5T^{2} \) |
| 11 | \( 1 + 4.24iT - 11T^{2} \) |
| 13 | \( 1 - 4.18iT - 13T^{2} \) |
| 17 | \( 1 + 2.44T + 17T^{2} \) |
| 19 | \( 1 - 4.89iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 10.2iT - 29T^{2} \) |
| 31 | \( 1 + 5.61iT - 31T^{2} \) |
| 37 | \( 1 + 3.24T + 37T^{2} \) |
| 41 | \( 1 - 2.44T + 41T^{2} \) |
| 43 | \( 1 - 7T + 43T^{2} \) |
| 47 | \( 1 - 7.94T + 47T^{2} \) |
| 53 | \( 1 - 2.48iT - 53T^{2} \) |
| 59 | \( 1 - 2.44T + 59T^{2} \) |
| 61 | \( 1 + 0.717iT - 61T^{2} \) |
| 67 | \( 1 + 3.48T + 67T^{2} \) |
| 71 | \( 1 + 12.7iT - 71T^{2} \) |
| 73 | \( 1 - 15.2iT - 73T^{2} \) |
| 79 | \( 1 - 9.24T + 79T^{2} \) |
| 83 | \( 1 - 5.49T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 + 6.63iT - 97T^{2} \) |
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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
−8.786584118778280466752278147764, −8.217805294395945863572529671041, −7.38480360634854530422813402289, −6.47503704390989137823946612155, −5.61505963344146660798263517602, −4.46856867805187817574716388128, −3.93259167266877535768275358844, −3.15701509156585692395755774083, −2.01974174182680185381827974692, −0.67743295333349314608845653816,
0.60200044865039844948310046568, 2.31698841400402936106761776552, 3.50309032284361117387764734757, 4.31971337715333045709445399927, 4.97911811642227979367760631030, 5.88199635042399400736052483266, 6.88144516773578204300151981079, 7.53961458770496460239298598781, 7.87329388612205511582013320469, 8.841358631090346140413508918864
