| L(s) = 1 | − 12·13-s + 24·31-s − 4·37-s + 8·61-s + 8·67-s + 36·73-s − 16·79-s + 28·97-s + 44·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 82·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 3.32·13-s + 4.31·31-s − 0.657·37-s + 1.02·61-s + 0.977·67-s + 4.21·73-s − 1.80·79-s + 2.84·97-s + 4.21·109-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.466332398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.466332398\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 206 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 994 T^{4} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 1918 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 104 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 4046 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 - 3998 T^{4} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 + 8722 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^3$ | \( 1 + 4322 T^{4} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.34616624160175317117862987412, −6.33926652332986210706872190765, −5.91410360291156811861583553259, −5.84996593310017166971536870797, −5.47104871635169696418280885024, −5.13644582759651307209736871374, −5.10217726810590971530042474673, −4.91645326347846769853672612901, −4.74065400831842140687790081279, −4.66974104326001042031070767239, −4.22798150948298739955702350555, −4.13744797566895491783551136216, −3.90850575028689153603819927014, −3.51333203036227568189253055405, −3.14110129035801284633297911549, −3.08399829351685933466716933406, −2.87160809641456649544138627828, −2.47261978423434892087305486331, −2.28497721499328196262736328232, −2.15799169824966320015913544463, −1.96907502518328916185172075376, −1.43225110954599236814652020384, −0.870072566747073790065392998315, −0.68223271897556611931616465208, −0.42329524986312848674696266031,
0.42329524986312848674696266031, 0.68223271897556611931616465208, 0.870072566747073790065392998315, 1.43225110954599236814652020384, 1.96907502518328916185172075376, 2.15799169824966320015913544463, 2.28497721499328196262736328232, 2.47261978423434892087305486331, 2.87160809641456649544138627828, 3.08399829351685933466716933406, 3.14110129035801284633297911549, 3.51333203036227568189253055405, 3.90850575028689153603819927014, 4.13744797566895491783551136216, 4.22798150948298739955702350555, 4.66974104326001042031070767239, 4.74065400831842140687790081279, 4.91645326347846769853672612901, 5.10217726810590971530042474673, 5.13644582759651307209736871374, 5.47104871635169696418280885024, 5.84996593310017166971536870797, 5.91410360291156811861583553259, 6.33926652332986210706872190765, 6.34616624160175317117862987412
