| L(s) = 1 | + 3-s + 1.08i·5-s − 2.42i·7-s + 9-s − i·11-s + (−2.23 − 2.83i)13-s + 1.08i·15-s + 1.16·17-s + 6.05i·19-s − 2.42i·21-s + 0.650·23-s + 3.82·25-s + 27-s + 0.993·29-s − 4.98i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.485i·5-s − 0.915i·7-s + 0.333·9-s − 0.301i·11-s + (−0.618 − 0.785i)13-s + 0.280i·15-s + 0.281·17-s + 1.38i·19-s − 0.528i·21-s + 0.135·23-s + 0.764·25-s + 0.192·27-s + 0.184·29-s − 0.894i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.177345565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.177345565\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (2.23 + 2.83i)T \) |
| good | 5 | \( 1 - 1.08iT - 5T^{2} \) |
| 7 | \( 1 + 2.42iT - 7T^{2} \) |
| 17 | \( 1 - 1.16T + 17T^{2} \) |
| 19 | \( 1 - 6.05iT - 19T^{2} \) |
| 23 | \( 1 - 0.650T + 23T^{2} \) |
| 29 | \( 1 - 0.993T + 29T^{2} \) |
| 31 | \( 1 + 4.98iT - 31T^{2} \) |
| 37 | \( 1 - 5.84iT - 37T^{2} \) |
| 41 | \( 1 + 9.99iT - 41T^{2} \) |
| 43 | \( 1 - 9.22T + 43T^{2} \) |
| 47 | \( 1 + 1.16iT - 47T^{2} \) |
| 53 | \( 1 + 8.43T + 53T^{2} \) |
| 59 | \( 1 + 7.90iT - 59T^{2} \) |
| 61 | \( 1 - 2.12T + 61T^{2} \) |
| 67 | \( 1 + 11.7iT - 67T^{2} \) |
| 71 | \( 1 + 13.7iT - 71T^{2} \) |
| 73 | \( 1 + 4.23iT - 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 0.514iT - 83T^{2} \) |
| 89 | \( 1 - 11.3iT - 89T^{2} \) |
| 97 | \( 1 + 15.3iT - 97T^{2} \) |
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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
−8.237998300173556061150494343502, −7.81224535355416185671343167779, −7.16420451021381072351471297308, −6.35502036378625033065809461214, −5.48484362737219047663254412402, −4.51575764286757874682410479142, −3.62495531135392062678306933134, −3.04688876315091100115207504662, −1.95555609297742365259568948829, −0.66945307523072899445079977163,
1.15390116448273413631959895469, 2.37113224124955552061446069038, 2.88673504234701940175582896842, 4.18899577539789136973123303699, 4.83411407852510705395247448655, 5.55159148019346329862772415805, 6.63495267812650513507778608539, 7.20825685245768791370257431926, 8.084946472763327919430584435352, 8.934008621558982904402970694948
