| L(s) = 1 | + (−1.22 + 1.22i)3-s − 2.44·5-s + (1 + 2.44i)7-s − 2.99i·9-s − 2.44i·13-s + (2.99 − 2.99i)15-s − 4.89·17-s − 2.44i·19-s + (−4.22 − 1.77i)21-s − 6i·23-s + 0.999·25-s + (3.67 + 3.67i)27-s + 6i·29-s + (−2.44 − 5.99i)35-s + 2·37-s + ⋯ |
| L(s) = 1 | + (−0.707 + 0.707i)3-s − 1.09·5-s + (0.377 + 0.925i)7-s − 0.999i·9-s − 0.679i·13-s + (0.774 − 0.774i)15-s − 1.18·17-s − 0.561i·19-s + (−0.921 − 0.387i)21-s − 1.25i·23-s + 0.199·25-s + (0.707 + 0.707i)27-s + 1.11i·29-s + (−0.414 − 1.01i)35-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7504822354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7504822354\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
| 7 | \( 1 + (-1 - 2.44i)T \) |
| good | 5 | \( 1 + 2.44T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 + 2.44iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 4.89T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 12.2iT - 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 9.79iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 2.44T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
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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
−9.481241516776044429590806128221, −8.761007830944461723965273949515, −8.123775954297019330792468059475, −7.01568126267459643881507341503, −6.19284776678851533764778476548, −5.18130993612111320894790478933, −4.55927359551235072689699061888, −3.63810058301232834793265164490, −2.47948362807726104950042591516, −0.45219101148943517851934378369,
0.929404761008709542750864254335, 2.23815481024385545411522403883, 3.94434655622340833293609367367, 4.36665652472949880544494838885, 5.55127204461544351173617560034, 6.53973916320152757045221116717, 7.35541669480826989514010200756, 7.74112112238123139826249501989, 8.620467373295022166664944385495, 9.768837083353615108442908293460
