| L(s) = 1 | + 4-s + 9-s + 6·11-s + 4·19-s + 6·29-s + 20·31-s + 36-s − 12·41-s + 6·44-s − 10·49-s − 18·59-s − 4·61-s − 64-s + 24·71-s + 4·76-s − 20·79-s − 36·89-s + 6·99-s − 12·101-s − 56·109-s + 6·116-s + 31·121-s + 20·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1/3·9-s + 1.80·11-s + 0.917·19-s + 1.11·29-s + 3.59·31-s + 1/6·36-s − 1.87·41-s + 0.904·44-s − 1.42·49-s − 2.34·59-s − 0.512·61-s − 1/8·64-s + 2.84·71-s + 0.458·76-s − 2.25·79-s − 3.81·89-s + 0.603·99-s − 1.19·101-s − 5.36·109-s + 0.557·116-s + 2.81·121-s + 1.79·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.961312175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.961312175\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^3$ | \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 25 T^{2} - 744 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 85 T^{2} + 5376 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 70 T^{2} + 411 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 12 T + 73 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 169 T^{2} + p^{2} T^{4} )( 1 + 167 T^{2} + p^{2} T^{4} ) \) |
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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
−6.72044078279408427916822209815, −6.45056273527734963874453129024, −6.03160302562207989389721136406, −5.96201138433748158462934882815, −5.79580795665875483145143530965, −5.63746582007997352059012868610, −5.08170376638841781803314581391, −5.00322981597346066446117497831, −4.91371659556913037704406045109, −4.45917400802815873378114971710, −4.27494701651679021953525465863, −4.22611763700011630604537169879, −4.17669054378569118983497200136, −3.45869110505010341893539207239, −3.38820413033894301381210582907, −3.10319032084578666643289212200, −3.08565411824742204712186507403, −2.72295003528674099123763671936, −2.29503920429047777819120637824, −2.22693088884867753732424398697, −1.64873438254111163943083053520, −1.27682277937301044327760350795, −1.18664031949751757219996932717, −1.15822960602034581788446562765, −0.26865949817725047723755134195,
0.26865949817725047723755134195, 1.15822960602034581788446562765, 1.18664031949751757219996932717, 1.27682277937301044327760350795, 1.64873438254111163943083053520, 2.22693088884867753732424398697, 2.29503920429047777819120637824, 2.72295003528674099123763671936, 3.08565411824742204712186507403, 3.10319032084578666643289212200, 3.38820413033894301381210582907, 3.45869110505010341893539207239, 4.17669054378569118983497200136, 4.22611763700011630604537169879, 4.27494701651679021953525465863, 4.45917400802815873378114971710, 4.91371659556913037704406045109, 5.00322981597346066446117497831, 5.08170376638841781803314581391, 5.63746582007997352059012868610, 5.79580795665875483145143530965, 5.96201138433748158462934882815, 6.03160302562207989389721136406, 6.45056273527734963874453129024, 6.72044078279408427916822209815
