| L(s) = 1 | − 4·4-s − 24·11-s + 16·16-s − 40·19-s + 60·29-s − 176·31-s − 84·41-s + 96·44-s + 430·49-s − 1.32e3·59-s − 1.07e3·61-s − 64·64-s − 1.58e3·71-s + 160·76-s + 1.04e3·79-s + 1.62e3·89-s + 1.23e3·101-s − 2.38e3·109-s − 240·116-s − 2.23e3·121-s + 704·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.657·11-s + 1/4·16-s − 0.482·19-s + 0.384·29-s − 1.01·31-s − 0.319·41-s + 0.328·44-s + 1.25·49-s − 2.91·59-s − 2.25·61-s − 1/8·64-s − 2.64·71-s + 0.241·76-s + 1.48·79-s + 1.92·89-s + 1.21·101-s − 2.09·109-s − 0.192·116-s − 1.67·121-s + 0.509·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7981200865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7981200865\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 430 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 12 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2950 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6050 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3890 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 30 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 88 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 36790 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 42 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 156310 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 198430 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 258550 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 660 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 538 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 179930 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 792 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 730510 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 520 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 901510 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 810 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 493630 T^{2} + p^{6} T^{4} \) |
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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
−10.72164211802323123963342068739, −10.39138656547957769057307234015, −10.27783486087779774072182538311, −9.273874378768512543731051823056, −9.139627339965092371262866722401, −8.869799873441265078387251335861, −8.055149003087572193962250993760, −7.63168844605255613402272413421, −7.53293204276345469382144785399, −6.49055607487871513053269022684, −6.37546034815046477423595667845, −5.52614459826089408960646001680, −5.28709740786007469243137774360, −4.41372057145090368654974775282, −4.32473750010219841107583361859, −3.32487699147239369143358851079, −2.95144416896806378516066210068, −2.06490353411434823870058777275, −1.36246324259157967577196335859, −0.29646943786919703730896526897,
0.29646943786919703730896526897, 1.36246324259157967577196335859, 2.06490353411434823870058777275, 2.95144416896806378516066210068, 3.32487699147239369143358851079, 4.32473750010219841107583361859, 4.41372057145090368654974775282, 5.28709740786007469243137774360, 5.52614459826089408960646001680, 6.37546034815046477423595667845, 6.49055607487871513053269022684, 7.53293204276345469382144785399, 7.63168844605255613402272413421, 8.055149003087572193962250993760, 8.869799873441265078387251335861, 9.139627339965092371262866722401, 9.273874378768512543731051823056, 10.27783486087779774072182538311, 10.39138656547957769057307234015, 10.72164211802323123963342068739
