| L(s) = 1 | + 2-s − 7-s − 8-s − 8·13-s − 14-s − 16-s + 6·17-s − 2·19-s − 3·23-s + 5·25-s − 8·26-s + 12·29-s − 5·31-s + 6·34-s − 8·37-s − 2·38-s − 6·41-s + 4·43-s − 3·46-s − 3·47-s − 6·49-s + 5·50-s + 6·53-s + 56-s + 12·58-s + 12·59-s − 8·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.377·7-s − 0.353·8-s − 2.21·13-s − 0.267·14-s − 1/4·16-s + 1.45·17-s − 0.458·19-s − 0.625·23-s + 25-s − 1.56·26-s + 2.22·29-s − 0.898·31-s + 1.02·34-s − 1.31·37-s − 0.324·38-s − 0.937·41-s + 0.609·43-s − 0.442·46-s − 0.437·47-s − 6/7·49-s + 0.707·50-s + 0.824·53-s + 0.133·56-s + 1.57·58-s + 1.56·59-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.807423923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.807423923\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.04032477864609268002565982447, −9.777515952474817901011434466223, −9.259877942501114268986310057630, −8.751326962335617125382252720620, −8.441946122149274835595135755070, −7.85635032229335404262425532342, −7.53145193190046028968492573059, −7.04761532862353707053308365296, −6.61847246084874769216130802483, −6.33389770815637281932093375757, −5.59831003165260407108373654232, −5.24531903110421178446444202967, −4.77164736425325512826385696324, −4.68535146355940493986559974537, −3.60953452347460093291097102409, −3.60713659143739222842826218370, −2.67843845010194044938166841293, −2.52521539617684817323654621671, −1.58962942518729663991156481580, −0.52050792294987665071416059564,
0.52050792294987665071416059564, 1.58962942518729663991156481580, 2.52521539617684817323654621671, 2.67843845010194044938166841293, 3.60713659143739222842826218370, 3.60953452347460093291097102409, 4.68535146355940493986559974537, 4.77164736425325512826385696324, 5.24531903110421178446444202967, 5.59831003165260407108373654232, 6.33389770815637281932093375757, 6.61847246084874769216130802483, 7.04761532862353707053308365296, 7.53145193190046028968492573059, 7.85635032229335404262425532342, 8.441946122149274835595135755070, 8.751326962335617125382252720620, 9.259877942501114268986310057630, 9.777515952474817901011434466223, 10.04032477864609268002565982447
