| L(s) = 1 | + 3-s − 4·7-s + 9-s − 2·11-s + 13-s + 4·17-s − 2·19-s − 4·21-s + 6·23-s + 27-s − 2·29-s + 4·31-s − 2·33-s − 6·37-s + 39-s − 6·41-s − 8·43-s − 8·47-s + 9·49-s + 4·51-s + 10·53-s − 2·57-s + 14·59-s + 10·61-s − 4·63-s − 4·67-s + 6·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 0.970·17-s − 0.458·19-s − 0.872·21-s + 1.25·23-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.348·33-s − 0.986·37-s + 0.160·39-s − 0.937·41-s − 1.21·43-s − 1.16·47-s + 9/7·49-s + 0.560·51-s + 1.37·53-s − 0.264·57-s + 1.82·59-s + 1.28·61-s − 0.503·63-s − 0.488·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−16.30568540965209, −15.70952778495319, −15.10235678522373, −14.81661918859279, −13.91593298435301, −13.41575667980567, −13.00977144930124, −12.59006310716126, −11.88478563781809, −11.22597371921718, −10.33407628595292, −10.02840597310218, −9.599810182440534, −8.656143274286704, −8.488098893865735, −7.592145182103129, −6.760022711382945, −6.696011241452095, −5.577897855850083, −5.169224258687182, −4.108646994461326, −3.382029605588135, −3.045130898884363, −2.207880269984157, −1.108802577701142, 0,
1.108802577701142, 2.207880269984157, 3.045130898884363, 3.382029605588135, 4.108646994461326, 5.169224258687182, 5.577897855850083, 6.696011241452095, 6.760022711382945, 7.592145182103129, 8.488098893865735, 8.656143274286704, 9.599810182440534, 10.02840597310218, 10.33407628595292, 11.22597371921718, 11.88478563781809, 12.59006310716126, 13.00977144930124, 13.41575667980567, 13.91593298435301, 14.81661918859279, 15.10235678522373, 15.70952778495319, 16.30568540965209
