| L(s) = 1 | + 6·3-s + 8·4-s + 27·9-s + 48·12-s + 48·16-s + 50·25-s + 108·27-s − 118·31-s + 216·36-s − 94·37-s + 288·48-s − 23·49-s + 256·64-s + 26·67-s + 300·75-s + 405·81-s − 708·93-s − 338·97-s + 400·100-s − 314·103-s + 864·108-s − 564·111-s − 944·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 2·3-s + 2·4-s + 3·9-s + 4·12-s + 3·16-s + 2·25-s + 4·27-s − 3.80·31-s + 6·36-s − 2.54·37-s + 6·48-s − 0.469·49-s + 4·64-s + 0.388·67-s + 4·75-s + 5·81-s − 7.61·93-s − 3.48·97-s + 4·100-s − 3.04·103-s + 8·108-s − 5.08·111-s − 7.61·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(8.936937780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.936937780\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 23 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 146 T^{2} + p^{4} T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 601 T^{2} + p^{4} T^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 59 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 47 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 3214 T^{2} + p^{4} T^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 7199 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 9791 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 12361 T^{2} + p^{4} T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 169 T + p^{2} T^{2} )^{2} \) |
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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
−11.13603096951249779300282852858, −10.95220924037691166144044296275, −10.55185551464710063165212461174, −10.17880938598833176776962579337, −9.402394231522816330464951057032, −9.216989537054446858937420532150, −8.447880683512287180381504150278, −8.306168937158940122416684308374, −7.50888762014276452804198175385, −7.27385793035487535678109372188, −6.74540057006090150455961510241, −6.72423690637824941369700314632, −5.49256043166584794127852791097, −5.21782623460569361829926393280, −4.03777506533911376239388590325, −3.52969422957625937247857782719, −3.07944644262946848786602618899, −2.54523945198002827748744763146, −1.74612591506220576899639675012, −1.50839786611026467460021403066,
1.50839786611026467460021403066, 1.74612591506220576899639675012, 2.54523945198002827748744763146, 3.07944644262946848786602618899, 3.52969422957625937247857782719, 4.03777506533911376239388590325, 5.21782623460569361829926393280, 5.49256043166584794127852791097, 6.72423690637824941369700314632, 6.74540057006090150455961510241, 7.27385793035487535678109372188, 7.50888762014276452804198175385, 8.306168937158940122416684308374, 8.447880683512287180381504150278, 9.216989537054446858937420532150, 9.402394231522816330464951057032, 10.17880938598833176776962579337, 10.55185551464710063165212461174, 10.95220924037691166144044296275, 11.13603096951249779300282852858
