| L(s) = 1 | + 8·3-s + 30·9-s + 16·19-s − 20·25-s + 40·27-s + 208·43-s − 172·49-s + 128·57-s − 112·67-s − 296·73-s − 160·75-s − 205·81-s − 248·97-s + 124·121-s + 127-s + 1.66e3·129-s + 131-s + 137-s + 139-s − 1.37e3·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 188·169-s + 480·171-s + ⋯ |
| L(s) = 1 | + 8/3·3-s + 10/3·9-s + 0.842·19-s − 4/5·25-s + 1.48·27-s + 4.83·43-s − 3.51·49-s + 2.24·57-s − 1.67·67-s − 4.05·73-s − 2.13·75-s − 2.53·81-s − 2.55·97-s + 1.02·121-s + 0.00787·127-s + 12.8·129-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 9.36·147-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.11·169-s + 2.80·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.124097829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.124097829\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 2 p T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{2}( 1 + 34 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1622 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 1334 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 1538 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 62 T + p^{2} T^{2} )^{2}( 1 + 62 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 52 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 578 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 2678 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 5342 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 7394 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 28 T + p^{2} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 9122 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 9782 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 802 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 12962 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 62 T + p^{2} T^{2} )^{4} \) |
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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
−7.35608708435318543356384756404, −7.13027090593379418387676349348, −6.99415613173825680593897566148, −6.58085746314872416460076645745, −6.17995463821952548108626671191, −6.11777028119355285466624444030, −5.67418169660150383217169614841, −5.65404835475930195871975302894, −5.50267958891979507265848568619, −4.81416898822747127361836529366, −4.77611053419152767233202773203, −4.32286458087558853265462802480, −4.18013661634736292627163672010, −3.89010249882778062225735856124, −3.88594457902502385822902356882, −3.23135923548350523650477121274, −2.95387210676326654573131396467, −2.90428337128448518754152546755, −2.89739548934757953577050728349, −2.37206738378480847053289582114, −1.93975759001326093416303431233, −1.64456499979261639760129387486, −1.53602975497591527181759416971, −0.896130968596955195092163490991, −0.30408412507495512913894672027,
0.30408412507495512913894672027, 0.896130968596955195092163490991, 1.53602975497591527181759416971, 1.64456499979261639760129387486, 1.93975759001326093416303431233, 2.37206738378480847053289582114, 2.89739548934757953577050728349, 2.90428337128448518754152546755, 2.95387210676326654573131396467, 3.23135923548350523650477121274, 3.88594457902502385822902356882, 3.89010249882778062225735856124, 4.18013661634736292627163672010, 4.32286458087558853265462802480, 4.77611053419152767233202773203, 4.81416898822747127361836529366, 5.50267958891979507265848568619, 5.65404835475930195871975302894, 5.67418169660150383217169614841, 6.11777028119355285466624444030, 6.17995463821952548108626671191, 6.58085746314872416460076645745, 6.99415613173825680593897566148, 7.13027090593379418387676349348, 7.35608708435318543356384756404
