| L(s) = 1 | + 4·3-s + 8·9-s − 10·25-s + 20·27-s − 36·43-s − 12·67-s − 40·75-s + 50·81-s − 44·83-s + 24·89-s − 52·107-s + 44·121-s + 127-s − 144·129-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 8/3·9-s − 2·25-s + 3.84·27-s − 5.48·43-s − 1.46·67-s − 4.61·75-s + 50/9·81-s − 4.82·83-s + 2.54·89-s − 5.02·107-s + 4·121-s + 0.0887·127-s − 12.6·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.457780710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.457780710\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.85928083917663559637149535568, −6.84216281421910205043302158550, −6.50991076542339294614602655947, −6.48243751937203348222984885297, −5.91431452623734054934344427457, −5.90944427046313956455996837439, −5.64599045946327585536656039542, −5.23238288962666012442365241136, −5.00249704699008140158491594392, −4.92228910750219418231042934010, −4.55538099017548746077301205700, −4.24979231299748181490956297812, −4.19474233711321209353980450553, −3.81826320582888585937437899577, −3.45258635464884659548061198255, −3.37545709014577787320360355877, −3.11518794983341993867974970416, −2.95435421094152390479808922316, −2.65512616103812100088702380052, −2.32394846213514521540316791353, −1.93986642201236842490104476465, −1.81925596462536633757577052229, −1.36153381813904542856434098732, −1.23334047248019587093103154412, −0.17450907632762951208522455336,
0.17450907632762951208522455336, 1.23334047248019587093103154412, 1.36153381813904542856434098732, 1.81925596462536633757577052229, 1.93986642201236842490104476465, 2.32394846213514521540316791353, 2.65512616103812100088702380052, 2.95435421094152390479808922316, 3.11518794983341993867974970416, 3.37545709014577787320360355877, 3.45258635464884659548061198255, 3.81826320582888585937437899577, 4.19474233711321209353980450553, 4.24979231299748181490956297812, 4.55538099017548746077301205700, 4.92228910750219418231042934010, 5.00249704699008140158491594392, 5.23238288962666012442365241136, 5.64599045946327585536656039542, 5.90944427046313956455996837439, 5.91431452623734054934344427457, 6.48243751937203348222984885297, 6.50991076542339294614602655947, 6.84216281421910205043302158550, 6.85928083917663559637149535568
