L(s) = 1 | − 2-s + 4-s − 8-s + 2·9-s + 6·11-s + 16-s − 2·18-s − 6·22-s + 8·23-s + 25-s − 32-s + 2·36-s + 4·37-s + 8·43-s + 6·44-s − 8·46-s − 50-s − 4·53-s + 64-s + 20·67-s + 20·71-s − 2·72-s − 4·74-s + 4·79-s − 5·81-s − 8·86-s − 6·88-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 2/3·9-s + 1.80·11-s + 1/4·16-s − 0.471·18-s − 1.27·22-s + 1.66·23-s + 1/5·25-s − 0.176·32-s + 1/3·36-s + 0.657·37-s + 1.21·43-s + 0.904·44-s − 1.17·46-s − 0.141·50-s − 0.549·53-s + 1/8·64-s + 2.44·67-s + 2.37·71-s − 0.235·72-s − 0.464·74-s + 0.450·79-s − 5/9·81-s − 0.862·86-s − 0.639·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.082141668\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.082141668\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 37 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
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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
−8.027366975273020026400887777178, −7.83793627640799082548581252165, −7.20879250232265376250088182274, −6.77820230933455951971846405361, −6.57782809010391781286002901101, −6.20107228903781990964333981045, −5.37292342895458086853336337282, −5.09700154424971948940316889393, −4.33293073680155005762941290420, −3.94640754065108654172353304781, −3.47494106061143408855061911123, −2.74429433532495952193708017562, −2.12208261364698785145198626275, −1.26440163312596409583792309937, −0.904606159115227586659809900858,
0.904606159115227586659809900858, 1.26440163312596409583792309937, 2.12208261364698785145198626275, 2.74429433532495952193708017562, 3.47494106061143408855061911123, 3.94640754065108654172353304781, 4.33293073680155005762941290420, 5.09700154424971948940316889393, 5.37292342895458086853336337282, 6.20107228903781990964333981045, 6.57782809010391781286002901101, 6.77820230933455951971846405361, 7.20879250232265376250088182274, 7.83793627640799082548581252165, 8.027366975273020026400887777178