L(s) = 1 | + 2.02i·2-s − i·3-s − 2.11·4-s + 2.02·6-s − 0.505i·7-s − 0.227i·8-s − 9-s + 0.687·11-s + 2.11i·12-s − 5.78i·13-s + 1.02·14-s − 3.76·16-s + 4.74i·17-s − 2.02i·18-s − 4.23·19-s + ⋯ |
L(s) = 1 | + 1.43i·2-s − 0.577i·3-s − 1.05·4-s + 0.827·6-s − 0.191i·7-s − 0.0806i·8-s − 0.333·9-s + 0.207·11-s + 0.609i·12-s − 1.60i·13-s + 0.274·14-s − 0.940·16-s + 1.15i·17-s − 0.477i·18-s − 0.972·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.004031751\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.004031751\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 2.02iT - 2T^{2} \) |
| 7 | \( 1 + 0.505iT - 7T^{2} \) |
| 11 | \( 1 - 0.687T + 11T^{2} \) |
| 13 | \( 1 + 5.78iT - 13T^{2} \) |
| 17 | \( 1 - 4.74iT - 17T^{2} \) |
| 19 | \( 1 + 4.23T + 19T^{2} \) |
| 23 | \( 1 - 8.36iT - 23T^{2} \) |
| 29 | \( 1 + 4.88T + 29T^{2} \) |
| 31 | \( 1 - 2.68T + 31T^{2} \) |
| 37 | \( 1 - 11.3iT - 37T^{2} \) |
| 41 | \( 1 - 0.144T + 41T^{2} \) |
| 43 | \( 1 - 5.94iT - 43T^{2} \) |
| 47 | \( 1 - 6.10iT - 47T^{2} \) |
| 53 | \( 1 - 10.8iT - 53T^{2} \) |
| 59 | \( 1 - 6.96T + 59T^{2} \) |
| 61 | \( 1 + 3.98T + 61T^{2} \) |
| 67 | \( 1 - 1.31iT - 67T^{2} \) |
| 71 | \( 1 + 3.79T + 71T^{2} \) |
| 73 | \( 1 + 10.9iT - 73T^{2} \) |
| 79 | \( 1 - 1.89T + 79T^{2} \) |
| 83 | \( 1 - 2.51iT - 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 + 1.17iT - 97T^{2} \) |
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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
−9.291056734787346956325727221673, −8.407604257901459345883909140949, −7.88404327371599445205798152943, −7.37514523465663029208900253470, −6.35954618441413385074056149948, −5.91929346545479135501717531435, −5.15022885492533785641662848461, −4.03626174816756832448057750198, −2.86044048859777189240794798993, −1.41430329008721056086877923041,
0.36395981354791073682087555253, 2.00954358860189099032557172816, 2.57718405594346176726575638977, 3.89992631796941372754639877692, 4.25927583529392579349307935819, 5.25259055070494721867925710373, 6.51082384228837292572085954748, 7.13868995214814275565550015670, 8.676976551912322029909153881161, 8.989791886369294790216752907003