| L(s) = 1 | − 3·2-s + 4-s − 7·7-s + 21·8-s + 60·11-s − 38·13-s + 21·14-s − 71·16-s + 84·17-s + 110·19-s − 180·22-s − 120·23-s + 114·26-s − 7·28-s + 162·29-s + 236·31-s + 45·32-s − 252·34-s + 376·37-s − 330·38-s − 126·41-s + 34·43-s + 60·44-s + 360·46-s + 6·47-s + 49·49-s − 38·52-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1/8·4-s − 0.377·7-s + 0.928·8-s + 1.64·11-s − 0.810·13-s + 0.400·14-s − 1.10·16-s + 1.19·17-s + 1.32·19-s − 1.74·22-s − 1.08·23-s + 0.859·26-s − 0.0472·28-s + 1.03·29-s + 1.36·31-s + 0.248·32-s − 1.27·34-s + 1.67·37-s − 1.40·38-s − 0.479·41-s + 0.120·43-s + 0.205·44-s + 1.15·46-s + 0.0186·47-s + 1/7·49-s − 0.101·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.259458145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.259458145\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 84 T + p^{3} T^{2} \) |
| 19 | \( 1 - 110 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 162 T + p^{3} T^{2} \) |
| 31 | \( 1 - 236 T + p^{3} T^{2} \) |
| 37 | \( 1 - 376 T + p^{3} T^{2} \) |
| 41 | \( 1 + 126 T + p^{3} T^{2} \) |
| 43 | \( 1 - 34 T + p^{3} T^{2} \) |
| 47 | \( 1 - 6 T + p^{3} T^{2} \) |
| 53 | \( 1 + 582 T + p^{3} T^{2} \) |
| 59 | \( 1 - 492 T + p^{3} T^{2} \) |
| 61 | \( 1 + 880 T + p^{3} T^{2} \) |
| 67 | \( 1 - 826 T + p^{3} T^{2} \) |
| 71 | \( 1 + 666 T + p^{3} T^{2} \) |
| 73 | \( 1 - 826 T + p^{3} T^{2} \) |
| 79 | \( 1 + 592 T + p^{3} T^{2} \) |
| 83 | \( 1 + 792 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1002 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1442 T + p^{3} T^{2} \) |
| 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
−9.275199666107111971597503930626, −8.262671905598554008194260425412, −7.69937519011113004098244554931, −6.83276582215055888044447485549, −6.02270053679299264650567972482, −4.84044872568776255138190729034, −3.99165220698615545959163756334, −2.88500847997515113332739930679, −1.45194209929227002936861618932, −0.71674928355056388130186062589,
0.71674928355056388130186062589, 1.45194209929227002936861618932, 2.88500847997515113332739930679, 3.99165220698615545959163756334, 4.84044872568776255138190729034, 6.02270053679299264650567972482, 6.83276582215055888044447485549, 7.69937519011113004098244554931, 8.262671905598554008194260425412, 9.275199666107111971597503930626
