| L(s) = 1 | + 6·3-s + 18·5-s + 27·9-s − 58·11-s + 48·13-s + 108·15-s − 6·17-s − 84·19-s − 6·23-s + 130·25-s + 108·27-s − 104·29-s − 252·31-s − 348·33-s + 224·37-s + 288·39-s + 438·41-s + 776·43-s + 486·45-s + 240·47-s − 36·51-s + 1.08e3·53-s − 1.04e3·55-s − 504·57-s − 312·59-s − 660·61-s + 864·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.60·5-s + 9-s − 1.58·11-s + 1.02·13-s + 1.85·15-s − 0.0856·17-s − 1.01·19-s − 0.0543·23-s + 1.03·25-s + 0.769·27-s − 0.665·29-s − 1.46·31-s − 1.83·33-s + 0.995·37-s + 1.18·39-s + 1.66·41-s + 2.75·43-s + 1.60·45-s + 0.744·47-s − 0.0988·51-s + 2.79·53-s − 2.55·55-s − 1.17·57-s − 0.688·59-s − 1.38·61-s + 1.64·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.528866900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.528866900\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 18 T + 194 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 58 T + 2270 T^{2} + 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 48 T + 2778 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 9698 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 84 T + 1782 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T - 6482 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 104 T + 46550 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 252 T + 74910 T^{2} + 252 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 224 T + 108918 T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 438 T + 113330 T^{2} - 438 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 388 T + p^{3} T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 240 T + 81758 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1080 T + 584422 T^{2} - 1080 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 312 T + 400022 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 660 T + 496554 T^{2} + 660 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 396 T + 399062 T^{2} - 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 66 T + 360574 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 972 T + 969842 T^{2} - 972 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 860 T + 1126590 T^{2} - 860 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2520 T + 2696102 T^{2} + 2520 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1686 T + 2089762 T^{2} - 1686 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2100 T + 2861538 T^{2} - 2100 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.881560446797156079801966532266, −8.738318990353735905656298600727, −7.890041417232435301536605048108, −7.81760025074929072264370962100, −7.31418063749322186238240207856, −7.16227694857730987845636546178, −6.27277263517035863369481384508, −6.05603364659633114838427010099, −5.72925217858204932895743878178, −5.53851373381353917707128931939, −4.72782190794944936840532925800, −4.50015044359038879456892995088, −3.74744236762429062186722687686, −3.65791206011254822425809517630, −2.77009233555504317668401669513, −2.50330456549303052910764198076, −2.02388374931356693390135508734, −1.97624827628271108411789854150, −0.947769038879111320623033531694, −0.60978641799910065497085969334,
0.60978641799910065497085969334, 0.947769038879111320623033531694, 1.97624827628271108411789854150, 2.02388374931356693390135508734, 2.50330456549303052910764198076, 2.77009233555504317668401669513, 3.65791206011254822425809517630, 3.74744236762429062186722687686, 4.50015044359038879456892995088, 4.72782190794944936840532925800, 5.53851373381353917707128931939, 5.72925217858204932895743878178, 6.05603364659633114838427010099, 6.27277263517035863369481384508, 7.16227694857730987845636546178, 7.31418063749322186238240207856, 7.81760025074929072264370962100, 7.890041417232435301536605048108, 8.738318990353735905656298600727, 8.881560446797156079801966532266
