| L(s) = 1 | − 4·4-s + 24·7-s − 48·13-s + 12·16-s + 24·19-s + 52·25-s − 96·28-s + 76·31-s − 44·37-s − 144·43-s + 170·49-s + 192·52-s + 336·61-s − 32·64-s − 140·67-s + 192·73-s − 96·76-s − 312·79-s − 1.15e3·91-s − 628·97-s − 208·100-s − 412·103-s + 288·112-s − 304·124-s + 127-s + 131-s + 576·133-s + ⋯ |
| L(s) = 1 | − 4-s + 24/7·7-s − 3.69·13-s + 3/4·16-s + 1.26·19-s + 2.07·25-s − 3.42·28-s + 2.45·31-s − 1.18·37-s − 3.34·43-s + 3.46·49-s + 3.69·52-s + 5.50·61-s − 1/2·64-s − 2.08·67-s + 2.63·73-s − 1.26·76-s − 3.94·79-s − 12.6·91-s − 6.47·97-s − 2.07·100-s − 4·103-s + 18/7·112-s − 2.45·124-s + 0.00787·127-s + 0.00763·131-s + 4.33·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.706893070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.706893070\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $D_{4}$ | \( ( 1 - 12 T + 131 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 + 24 T + 374 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 362 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 12 T + 251 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1168 T^{2} + 857538 T^{4} - 1168 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1816 T^{2} + 1679154 T^{4} - 1816 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 38 T + 2175 T^{2} - 38 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 22 T + 1131 T^{2} + 22 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 1288 T^{2} + 5304210 T^{4} - 1288 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 72 T + 4022 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 7072 T^{2} + 10050 p^{2} T^{4} - 7072 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 8608 T^{2} + 32750178 T^{4} - 8608 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1168 T^{2} + 105570 T^{4} - 1168 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 168 T + 14423 T^{2} - 168 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 70 T + 4911 T^{2} + 70 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 19540 T^{2} + 146248614 T^{4} - 19540 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 96 T + 11879 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 156 T + 17699 T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 88 T^{2} - 93260622 T^{4} - 88 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 14212 T^{2} + 173484486 T^{4} - 14212 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 157 T + p^{2} T^{2} )^{4} \) |
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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
−6.51788055989759225083541646478, −5.68994743453275720213929765461, −5.48505093570208442718526505188, −5.43649439534158023074465673311, −5.43557453370894057131785616336, −5.00373403941848695071657916855, −4.92103111659303846214720616611, −4.87091945578879226802871920194, −4.85310160448897351425398829860, −4.32789779941882144862416461957, −4.18020053070680596429323551411, −4.13119221831281195522085479899, −3.92573748483278994527709254692, −3.11572721926793210273007188755, −2.98499641913814051578330046907, −2.91670794450220713934494322859, −2.85899656193795513562558602862, −2.37096485048333367647947922109, −1.94774561568431317399401961223, −1.73786430351044134878753424461, −1.65726475509605953516492095471, −1.27506697926658778619393941792, −0.978085422736422978245943247718, −0.49803572965983433881666009230, −0.32297895357631536346402407215,
0.32297895357631536346402407215, 0.49803572965983433881666009230, 0.978085422736422978245943247718, 1.27506697926658778619393941792, 1.65726475509605953516492095471, 1.73786430351044134878753424461, 1.94774561568431317399401961223, 2.37096485048333367647947922109, 2.85899656193795513562558602862, 2.91670794450220713934494322859, 2.98499641913814051578330046907, 3.11572721926793210273007188755, 3.92573748483278994527709254692, 4.13119221831281195522085479899, 4.18020053070680596429323551411, 4.32789779941882144862416461957, 4.85310160448897351425398829860, 4.87091945578879226802871920194, 4.92103111659303846214720616611, 5.00373403941848695071657916855, 5.43557453370894057131785616336, 5.43649439534158023074465673311, 5.48505093570208442718526505188, 5.68994743453275720213929765461, 6.51788055989759225083541646478
