| L(s) = 1 | + 100·25-s + 188·49-s + 184·73-s + 8·97-s − 484·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 292·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 4·25-s + 3.83·49-s + 2.52·73-s + 8/97·97-s − 4·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.72·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.744653379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.744653379\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2}( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 194 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2}( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 3214 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )^{2}( 1 + 74 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 5906 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 7682 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{4} \) |
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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
−7.65355876456543215151201543778, −7.02490504791265514521467697435, −6.98948432355334037837907567891, −6.96773652281922726394799692837, −6.62535053715603476380293769815, −6.43224725911316604670320885829, −6.02975636525288714072338045297, −5.73010845996528658750122603362, −5.67993148568713205382621596783, −5.12323943451003143085071515150, −5.06792399732378538524718140721, −4.89422410287713602516209439877, −4.58073334881457767630510195051, −4.23002205156934276259931017362, −3.91130270336800503448497964500, −3.63504567199759858276862580193, −3.48732582742696700035700431428, −2.89784795359961233413665776440, −2.62726926302306247602268130861, −2.60624224598161894703331059898, −2.18015835397173476993150923109, −1.58036866726169229019531761655, −1.08994248345608939333191512361, −0.911876019879196787135317214468, −0.44926220526481729340342506136,
0.44926220526481729340342506136, 0.911876019879196787135317214468, 1.08994248345608939333191512361, 1.58036866726169229019531761655, 2.18015835397173476993150923109, 2.60624224598161894703331059898, 2.62726926302306247602268130861, 2.89784795359961233413665776440, 3.48732582742696700035700431428, 3.63504567199759858276862580193, 3.91130270336800503448497964500, 4.23002205156934276259931017362, 4.58073334881457767630510195051, 4.89422410287713602516209439877, 5.06792399732378538524718140721, 5.12323943451003143085071515150, 5.67993148568713205382621596783, 5.73010845996528658750122603362, 6.02975636525288714072338045297, 6.43224725911316604670320885829, 6.62535053715603476380293769815, 6.96773652281922726394799692837, 6.98948432355334037837907567891, 7.02490504791265514521467697435, 7.65355876456543215151201543778
