| L(s) = 1 | + i·3-s − 2.08i·5-s − 5.03·7-s − 9-s + 0.828i·11-s − 2.94i·13-s + 2.08·15-s + 4.82·17-s + 2.82i·19-s − 5.03i·21-s + 4.16·23-s + 0.656·25-s − i·27-s + 7.97i·29-s − 5.03·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.932i·5-s − 1.90·7-s − 0.333·9-s + 0.249i·11-s − 0.817i·13-s + 0.538·15-s + 1.17·17-s + 0.648i·19-s − 1.09i·21-s + 0.869·23-s + 0.131·25-s − 0.192i·27-s + 1.48i·29-s − 0.903·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8955255001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8955255001\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| good | 5 | \( 1 + 2.08iT - 5T^{2} \) |
| 7 | \( 1 + 5.03T + 7T^{2} \) |
| 11 | \( 1 - 0.828iT - 11T^{2} \) |
| 13 | \( 1 + 2.94iT - 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 - 4.16T + 23T^{2} \) |
| 29 | \( 1 - 7.97iT - 29T^{2} \) |
| 31 | \( 1 + 5.03T + 31T^{2} \) |
| 37 | \( 1 - 7.11iT - 37T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 - 12.4iT - 43T^{2} \) |
| 47 | \( 1 + 4.16T + 47T^{2} \) |
| 53 | \( 1 - 12.1iT - 53T^{2} \) |
| 59 | \( 1 - 1.65iT - 59T^{2} \) |
| 61 | \( 1 + 7.11iT - 61T^{2} \) |
| 67 | \( 1 + 2.34iT - 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 5.03T + 79T^{2} \) |
| 83 | \( 1 - 3.17iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 0.343T + 97T^{2} \) |
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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
−9.704444289263313864972565990940, −9.028774201704728020891581236520, −8.243211886416713785611752375986, −7.21495254331663949247926892881, −6.29128759200733771115672863928, −5.47336933849838659194912454525, −4.75218968380382044098501770858, −3.40145011658186797323700452659, −3.12414640367897992340790257158, −1.13066910219665962873363184872,
0.39766115295678557302511248842, 2.23808516812216536091062178523, 3.17920525369256454144286557693, 3.75799880549871706845291518964, 5.40051898406949006194647819583, 6.21724056543293806397901573500, 6.96533103598863319052788826474, 7.19857064359044888937197794809, 8.522394992212950350766449518068, 9.363853196018724319196151715992
