| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 2·8-s + 9-s − 4·11-s − 6·12-s − 4·17-s − 2·18-s + 8·22-s + 8·23-s + 4·24-s − 4·25-s + 2·27-s − 4·29-s + 16·31-s + 6·32-s + 8·33-s + 8·34-s + 3·36-s − 4·37-s + 16·41-s − 8·43-s − 12·44-s − 16·46-s + 24·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 0.707·8-s + 1/3·9-s − 1.20·11-s − 1.73·12-s − 0.970·17-s − 0.471·18-s + 1.70·22-s + 1.66·23-s + 0.816·24-s − 4/5·25-s + 0.384·27-s − 0.742·29-s + 2.87·31-s + 1.06·32-s + 1.39·33-s + 1.37·34-s + 1/2·36-s − 0.657·37-s + 2.49·41-s − 1.21·43-s − 1.80·44-s − 2.35·46-s + 3.50·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6770926459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6770926459\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p T + T^{2} - p T^{3} - 3 T^{4} - p^{2} T^{5} + p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^3$ | \( 1 - 6 T^{2} - 13 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_4\times C_2$ | \( 1 + 4 T + 10 T^{2} - 112 T^{3} - 525 T^{4} - 112 p T^{5} + 10 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T - 30 T^{2} - 112 T^{3} + 155 T^{4} - 112 p T^{5} - 30 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 16 T + 118 T^{2} - 896 T^{3} + 6867 T^{4} - 896 p T^{5} + 118 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T - 6 T^{2} - 128 T^{3} + 299 T^{4} - 128 p T^{5} - 6 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 - 4 T - 74 T^{2} + 112 T^{3} + 3675 T^{4} + 112 p T^{5} - 74 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 4 T + 18 T^{2} - 496 T^{3} - 4693 T^{4} - 496 p T^{5} + 18 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T - 78 T^{2} - 64 T^{3} + 11387 T^{4} - 64 p T^{5} - 78 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 2 T - 67 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 24 T + 262 T^{2} - 3264 T^{3} + 39411 T^{4} - 3264 p T^{5} + 262 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 4 T - 150 T^{2} - 112 T^{3} + 16595 T^{4} - 112 p T^{5} - 150 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.85367015335727319243938035844, −7.63766527649877479335388198009, −7.44887170621006871566221752535, −7.29809213740979468354591257644, −6.90616231386214811538285367320, −6.78441309465247104042041703103, −6.40940494951528531770607497376, −6.24109139264742190503335946828, −6.00879206887661471875555171306, −5.69883922840257998416938983799, −5.54158196755666875524205763952, −5.25187016563718318674416825070, −4.79152638643166828516196947337, −4.54936766426418885587065903167, −4.53521804362827743023511464805, −4.21130956263690406268950429095, −3.61134353718673147896807893578, −3.33327621237102495705815664279, −2.79537479818816027185796574490, −2.68088528151026880490290176226, −2.20169886619449693510770836419, −2.12457780496368629260633669199, −1.30196099758597099383030985737, −0.878569479498000340097638638849, −0.51156222768782491682817702019,
0.51156222768782491682817702019, 0.878569479498000340097638638849, 1.30196099758597099383030985737, 2.12457780496368629260633669199, 2.20169886619449693510770836419, 2.68088528151026880490290176226, 2.79537479818816027185796574490, 3.33327621237102495705815664279, 3.61134353718673147896807893578, 4.21130956263690406268950429095, 4.53521804362827743023511464805, 4.54936766426418885587065903167, 4.79152638643166828516196947337, 5.25187016563718318674416825070, 5.54158196755666875524205763952, 5.69883922840257998416938983799, 6.00879206887661471875555171306, 6.24109139264742190503335946828, 6.40940494951528531770607497376, 6.78441309465247104042041703103, 6.90616231386214811538285367320, 7.29809213740979468354591257644, 7.44887170621006871566221752535, 7.63766527649877479335388198009, 7.85367015335727319243938035844
