| L(s) = 1 | + 2-s − 2·3-s + 4-s + 5-s − 2·6-s + 9-s + 10-s + 11-s − 2·12-s − 2·15-s − 4·17-s + 18-s + 20-s + 22-s + 25-s + 29-s − 2·30-s + 31-s − 32-s − 2·33-s − 4·34-s + 36-s + 44-s + 45-s + 4·49-s + 50-s + 8·51-s + ⋯ |
| L(s) = 1 | + 2-s − 2·3-s + 4-s + 5-s − 2·6-s + 9-s + 10-s + 11-s − 2·12-s − 2·15-s − 4·17-s + 18-s + 20-s + 22-s + 25-s + 29-s − 2·30-s + 31-s − 32-s − 2·33-s − 4·34-s + 36-s + 44-s + 45-s + 4·49-s + 50-s + 8·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(239^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(239^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2943858106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2943858106\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 239 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 3 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 5 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 11 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 71 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
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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
−9.073238867741447057898324776284, −8.981340611610185972617487201368, −8.880106534564560020784796588050, −8.506379133538243806205135258458, −7.898247796424789213962926104483, −7.76982960953104650714598932885, −7.23908620515556811794894470155, −6.96859303265943508864605154875, −6.74313811730359380167782410744, −6.63628709256952529502333976324, −6.24978106129836996910565784969, −6.03386551036409625280310185267, −5.93744985048905186143138466949, −5.71754835930322359626052258075, −5.39468122066492703540558794955, −5.05871805908096797741426468348, −4.53617272129562742363437552652, −4.39927094325361724666637173768, −4.27753201481372775171416648494, −3.93309759833356519060862370172, −3.01044751429710392814947586285, −2.83927232801964389583926020804, −2.63560185139595457615535339359, −1.90306220356211129036976543826, −1.54432900565373525411282010947,
1.54432900565373525411282010947, 1.90306220356211129036976543826, 2.63560185139595457615535339359, 2.83927232801964389583926020804, 3.01044751429710392814947586285, 3.93309759833356519060862370172, 4.27753201481372775171416648494, 4.39927094325361724666637173768, 4.53617272129562742363437552652, 5.05871805908096797741426468348, 5.39468122066492703540558794955, 5.71754835930322359626052258075, 5.93744985048905186143138466949, 6.03386551036409625280310185267, 6.24978106129836996910565784969, 6.63628709256952529502333976324, 6.74313811730359380167782410744, 6.96859303265943508864605154875, 7.23908620515556811794894470155, 7.76982960953104650714598932885, 7.898247796424789213962926104483, 8.506379133538243806205135258458, 8.880106534564560020784796588050, 8.981340611610185972617487201368, 9.073238867741447057898324776284
