| L(s) = 1 | + 2·2-s − 4-s − 8·8-s − 6·9-s − 2·11-s − 7·16-s − 12·18-s − 4·22-s + 8·23-s + 25-s + 12·29-s + 14·32-s + 6·36-s − 4·37-s + 8·43-s + 2·44-s + 16·46-s − 7·49-s + 2·50-s − 4·53-s + 24·58-s + 35·64-s − 32·67-s + 16·71-s + 48·72-s − 8·74-s + 16·79-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s − 2.82·8-s − 2·9-s − 0.603·11-s − 7/4·16-s − 2.82·18-s − 0.852·22-s + 1.66·23-s + 1/5·25-s + 2.22·29-s + 2.47·32-s + 36-s − 0.657·37-s + 1.21·43-s + 0.301·44-s + 2.35·46-s − 49-s + 0.282·50-s − 0.549·53-s + 3.15·58-s + 35/8·64-s − 3.90·67-s + 1.89·71-s + 5.65·72-s − 0.929·74-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 148225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.338464506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.338464506\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.365267708430525132762162914792, −8.781700662013046394929624444870, −8.248642768475844318802620430882, −8.240174245715852192939095672164, −7.24720948442492513050045074918, −6.40684005531684016854972661238, −6.16821463973303244938343944630, −5.57003540112200356703114172743, −5.09766059609940450216711100862, −4.82709630451403100091257393731, −4.26300768017318841213602111936, −3.25423179226253980082861279896, −3.10898016437346903231946450096, −2.59341539336063966352674092067, −0.61786367914774168667137108260,
0.61786367914774168667137108260, 2.59341539336063966352674092067, 3.10898016437346903231946450096, 3.25423179226253980082861279896, 4.26300768017318841213602111936, 4.82709630451403100091257393731, 5.09766059609940450216711100862, 5.57003540112200356703114172743, 6.16821463973303244938343944630, 6.40684005531684016854972661238, 7.24720948442492513050045074918, 8.240174245715852192939095672164, 8.248642768475844318802620430882, 8.781700662013046394929624444870, 9.365267708430525132762162914792
