| L(s) = 1 | + 7-s + 2·11-s − 2·13-s − 6·17-s − 4·19-s + 4·23-s − 10·29-s + 6·31-s + 10·37-s + 2·41-s − 6·43-s + 6·47-s + 49-s + 2·53-s + 12·59-s − 4·61-s + 14·67-s − 4·71-s + 2·73-s + 2·77-s − 8·79-s − 4·83-s − 6·89-s − 2·91-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.603·11-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 0.834·23-s − 1.85·29-s + 1.07·31-s + 1.64·37-s + 0.312·41-s − 0.914·43-s + 0.875·47-s + 1/7·49-s + 0.274·53-s + 1.56·59-s − 0.512·61-s + 1.71·67-s − 0.474·71-s + 0.234·73-s + 0.227·77-s − 0.900·79-s − 0.439·83-s − 0.635·89-s − 0.209·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.85232172120162, −15.97678949388721, −15.33156896695810, −14.82866265538673, −14.57869948718269, −13.67271977439231, −13.13864491537485, −12.78015955425879, −11.92521169346159, −11.36539049366044, −11.00882338936840, −10.30998777757747, −9.469647800888933, −9.144368157667114, −8.393275501558430, −7.871311183347580, −6.917578962357624, −6.705856527893205, −5.789854520363333, −5.116541727051978, −4.260270997760910, −3.992262830877398, −2.718776307113659, −2.195061270523617, −1.198408836215801, 0,
1.198408836215801, 2.195061270523617, 2.718776307113659, 3.992262830877398, 4.260270997760910, 5.116541727051978, 5.789854520363333, 6.705856527893205, 6.917578962357624, 7.871311183347580, 8.393275501558430, 9.144368157667114, 9.469647800888933, 10.30998777757747, 11.00882338936840, 11.36539049366044, 11.92521169346159, 12.78015955425879, 13.13864491537485, 13.67271977439231, 14.57869948718269, 14.82866265538673, 15.33156896695810, 15.97678949388721, 16.85232172120162
