| L(s) = 1 | + 9-s + 2·11-s + 2·29-s − 49-s − 4·71-s − 2·79-s + 2·99-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 9-s + 2·11-s + 2·29-s − 49-s − 4·71-s − 2·79-s + 2·99-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.808146303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.808146303\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_1$ | \( ( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.026368704389947980846614542107, −8.763040401577450517067061640054, −8.719092726439537369148181610955, −7.83577241949867935734164701876, −7.82013759181595875966845514641, −7.15288482848796159652818718685, −6.76720951016542946468186860957, −6.73690912744666326733968084455, −6.05700980466557834257382444348, −5.97417910944711007300441616314, −5.28996130437705788064625617305, −4.66904713183137169292203587462, −4.42400484698112915864487358015, −4.17542394167422017363378338004, −3.66365492791129336934090479355, −3.02910236435437842058177517255, −2.80607882101970589734710714815, −1.80012982497458135385205880478, −1.51528762134580811113057519455, −0.980421522165142962586538581285,
0.980421522165142962586538581285, 1.51528762134580811113057519455, 1.80012982497458135385205880478, 2.80607882101970589734710714815, 3.02910236435437842058177517255, 3.66365492791129336934090479355, 4.17542394167422017363378338004, 4.42400484698112915864487358015, 4.66904713183137169292203587462, 5.28996130437705788064625617305, 5.97417910944711007300441616314, 6.05700980466557834257382444348, 6.73690912744666326733968084455, 6.76720951016542946468186860957, 7.15288482848796159652818718685, 7.82013759181595875966845514641, 7.83577241949867935734164701876, 8.719092726439537369148181610955, 8.763040401577450517067061640054, 9.026368704389947980846614542107
