L(s) = 1 | − i·2-s − 2i·3-s − 4-s − 2·6-s − i·7-s + i·8-s − 9-s − 11-s + 2i·12-s + 2i·13-s − 14-s + 16-s − 6i·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 − 0.377i·7-s + 0.353i·8-s − 0.333·9-s − 0.301·11-s + 0.577i·12-s + 0.554i·13-s − 0.267·14-s + 0.250·16-s − 1.45i·17-s + 0.235i·18-s − 0.458·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.056582103\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.056582103\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 18T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−7.897894457760361382740339582380, −7.29935467567606239808091724305, −6.64528148177060704380301008250, −5.87955401896915691480635373505, −4.71980815172921511580929961786, −4.21755996169935317041061406748, −2.87810699699757592728356600876, −2.31580726865117764745013998178, −1.25194369111952391920379542501, −0.32414433511706575374543522213,
1.54959607914988530110163358505, 3.03290077045981257801998467595, 3.73134471099496489293692804294, 4.62626840556589829670061105187, 5.12961404772351430051208950385, 6.03739051632072380422913906033, 6.53698197148980954081587392981, 7.81840832332704904083289543596, 8.110100837725497224495687874140, 9.042254935673947115081976576770