| L(s) = 1 | − i·2-s + 2i·3-s − 4-s + 2·6-s + 4.37i·7-s + i·8-s − 9-s + 2.37·11-s − 2i·12-s + 6.74i·13-s + 4.37·14-s + 16-s − 0.372i·17-s + i·18-s + 2·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 1.15i·3-s − 0.5·4-s + 0.816·6-s + 1.65i·7-s + 0.353i·8-s − 0.333·9-s + 0.715·11-s − 0.577i·12-s + 1.87i·13-s + 1.16·14-s + 0.250·16-s − 0.0902i·17-s + 0.235i·18-s + 0.458·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.575070771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.575070771\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 37 | \( 1 + iT \) |
| good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 7 | \( 1 - 4.37iT - 7T^{2} \) |
| 11 | \( 1 - 2.37T + 11T^{2} \) |
| 13 | \( 1 - 6.74iT - 13T^{2} \) |
| 17 | \( 1 + 0.372iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 4.74iT - 23T^{2} \) |
| 29 | \( 1 - 9.11T + 29T^{2} \) |
| 31 | \( 1 + 8.37T + 31T^{2} \) |
| 41 | \( 1 + 0.372T + 41T^{2} \) |
| 43 | \( 1 - 1.62iT - 43T^{2} \) |
| 47 | \( 1 + 2.74iT - 47T^{2} \) |
| 53 | \( 1 + 4.37iT - 53T^{2} \) |
| 59 | \( 1 + 1.25T + 59T^{2} \) |
| 61 | \( 1 - 0.372T + 61T^{2} \) |
| 67 | \( 1 - 6.74iT - 67T^{2} \) |
| 71 | \( 1 - 4.74T + 71T^{2} \) |
| 73 | \( 1 + 2.74iT - 73T^{2} \) |
| 79 | \( 1 + 6.74T + 79T^{2} \) |
| 83 | \( 1 - 10.7iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 17.1iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.514806772673091842066333107225, −8.882457505763335356096428870909, −8.584767947445242419790507589658, −7.00642997768412131587643184031, −6.14338762311085974003707036347, −5.14359705671449803779097246134, −4.49652811124892316417028724592, −3.71203302460700589866617775159, −2.65032124187323802949963671772, −1.69980309501955514649945022663,
0.64101568177406071564906113649, 1.38623794677194177544041518889, 3.14596098301619792561145915513, 4.00314964155926738473941203724, 5.08049194951904396995340977804, 6.05221953030667607155144760329, 6.77966797058911990881403398093, 7.59272952095495634395613764824, 7.66854360324189428447016652469, 8.678679888277732134010284925097
