| L(s) = 1 | − 2.50e3·25-s − 4.98e3·41-s + 9.60e3·49-s + 1.19e4·81-s + 6.37e4·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.14e5·169-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 − 2.96·41-s + 4·49-s + 1.82·81-s + 4.99·113-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s − 4·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + 2.24e−5·211-s + 2.01e−5·223-s + 1.94e−5·227-s + 1.90e−5·229-s + 1.84e−5·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.843872565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.843872565\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T + 128 T^{2} - 16 p^{4} T^{3} + p^{8} T^{4} )( 1 + 16 T + 128 T^{2} + 16 p^{4} T^{3} + p^{8} T^{4} ) \) |
| 5 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 336 T + 56448 T^{2} - 336 p^{4} T^{3} + p^{8} T^{4} )( 1 + 336 T + 56448 T^{2} + 336 p^{4} T^{3} + p^{8} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 162434 T^{2} + p^{8} T^{4} )^{2} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 816 T + 332928 T^{2} - 816 p^{4} T^{3} + p^{8} T^{4} )( 1 + 816 T + 332928 T^{2} + 816 p^{4} T^{3} + p^{8} T^{4} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + 1246 T + p^{4} T^{2} )^{4} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 1680 T + 1411200 T^{2} - 1680 p^{4} T^{3} + p^{8} T^{4} )( 1 + 1680 T + 1411200 T^{2} + 1680 p^{4} T^{3} + p^{8} T^{4} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 9840 T + 48412800 T^{2} - 9840 p^{4} T^{3} + p^{8} T^{4} )( 1 + 9840 T + 48412800 T^{2} + 9840 p^{4} T^{3} + p^{8} T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 10416 T + 54246528 T^{2} - 10416 p^{4} T^{3} + p^{8} T^{4} )( 1 + 10416 T + 54246528 T^{2} + 10416 p^{4} T^{3} + p^{8} T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 33567554 T^{2} + p^{8} T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 11376 T + 64706688 T^{2} - 11376 p^{4} T^{3} + p^{8} T^{4} )( 1 + 11376 T + 64706688 T^{2} + 11376 p^{4} T^{3} + p^{8} T^{4} ) \) |
| 89 | $C_2^2$ | \( ( 1 - 95519806 T^{2} + p^{8} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 77418238 T^{2} + p^{8} T^{4} )^{2} \) |
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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
−7.34859953932671051893093665352, −7.04394124340997856009070807259, −6.81880618778870393492693167947, −6.37940750262000992600666076253, −6.16944980393294970754882365240, −5.97997689710996436107385048530, −5.95768999728284566213097079613, −5.36172906379664319431801598052, −5.26164982384586200167703355054, −5.18223931146701637780044367077, −4.71977517129024375456108553648, −4.24250259937419065413060673091, −4.20766943833263545566981570874, −3.90186913389530311342874067611, −3.58579307692419175647315525131, −3.28352451673094696434842355656, −3.27780487174741480654897359254, −2.38807634330955478995553584981, −2.33790977264532677063234674623, −2.12564228329490276470349369642, −1.82077455348637274581402495920, −1.29081953406011910443634915290, −1.05881031134921103091271798068, −0.41749731806350865706967599195, −0.24667328690646705761275644083,
0.24667328690646705761275644083, 0.41749731806350865706967599195, 1.05881031134921103091271798068, 1.29081953406011910443634915290, 1.82077455348637274581402495920, 2.12564228329490276470349369642, 2.33790977264532677063234674623, 2.38807634330955478995553584981, 3.27780487174741480654897359254, 3.28352451673094696434842355656, 3.58579307692419175647315525131, 3.90186913389530311342874067611, 4.20766943833263545566981570874, 4.24250259937419065413060673091, 4.71977517129024375456108553648, 5.18223931146701637780044367077, 5.26164982384586200167703355054, 5.36172906379664319431801598052, 5.95768999728284566213097079613, 5.97997689710996436107385048530, 6.16944980393294970754882365240, 6.37940750262000992600666076253, 6.81880618778870393492693167947, 7.04394124340997856009070807259, 7.34859953932671051893093665352
