| L(s) = 1 | + 1.78i·2-s − 0.213i·3-s − 1.17·4-s + (2.04 + 0.913i)5-s + 0.380·6-s + 3.50i·7-s + 1.47i·8-s + 2.95·9-s + (−1.62 + 3.63i)10-s − 11-s + 0.250i·12-s + 1.11i·13-s − 6.25·14-s + (0.194 − 0.435i)15-s − 4.97·16-s − 3.28i·17-s + ⋯ |
| L(s) = 1 | + 1.25i·2-s − 0.123i·3-s − 0.585·4-s + (0.912 + 0.408i)5-s + 0.155·6-s + 1.32i·7-s + 0.521i·8-s + 0.984·9-s + (−0.514 + 1.14i)10-s − 0.301·11-s + 0.0722i·12-s + 0.309i·13-s − 1.67·14-s + (0.0503 − 0.112i)15-s − 1.24·16-s − 0.795i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.032554299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.032554299\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.04 - 0.913i)T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.78iT - 2T^{2} \) |
| 3 | \( 1 + 0.213iT - 3T^{2} \) |
| 7 | \( 1 - 3.50iT - 7T^{2} \) |
| 13 | \( 1 - 1.11iT - 13T^{2} \) |
| 17 | \( 1 + 3.28iT - 17T^{2} \) |
| 23 | \( 1 + 0.303iT - 23T^{2} \) |
| 29 | \( 1 - 5.23T + 29T^{2} \) |
| 31 | \( 1 + 0.126T + 31T^{2} \) |
| 37 | \( 1 + 5.03iT - 37T^{2} \) |
| 41 | \( 1 - 0.293T + 41T^{2} \) |
| 43 | \( 1 + 0.180iT - 43T^{2} \) |
| 47 | \( 1 + 3.64iT - 47T^{2} \) |
| 53 | \( 1 - 1.17iT - 53T^{2} \) |
| 59 | \( 1 + 5.73T + 59T^{2} \) |
| 61 | \( 1 + 3.61T + 61T^{2} \) |
| 67 | \( 1 + 5.70iT - 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 12.3iT - 73T^{2} \) |
| 79 | \( 1 - 2.32T + 79T^{2} \) |
| 83 | \( 1 - 2.66iT - 83T^{2} \) |
| 89 | \( 1 + 0.0652T + 89T^{2} \) |
| 97 | \( 1 + 17.5iT - 97T^{2} \) |
| 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
−10.02972086006920740201121620477, −9.222714968545728142696439626239, −8.596926783064493949125761578161, −7.54008332543175198341043142396, −6.84017845944845527197880495922, −6.12782877995760269388416501190, −5.42812740633646047410323480171, −4.61307707254554744512886020458, −2.78850185781383345974594097966, −1.93382256473730150170184533965,
0.968633255006608384086507968799, 1.78936317226262983416203547024, 3.08103309610683137972856784256, 4.15298713810587003531448950970, 4.76914530118479423164325052223, 6.20288472008407768212910437136, 6.99472957591024393517753309739, 7.994640154924216627759788932330, 9.153746321364094925638831171961, 9.951600120193670135951840830688
