| L(s) = 1 | − 2·4-s + 3·16-s + 20·19-s − 10·25-s + 12·29-s − 16·31-s − 32·41-s + 16·49-s − 16·59-s − 8·61-s − 4·64-s − 8·71-s − 40·76-s + 16·79-s − 16·89-s + 20·100-s + 28·101-s + 36·109-s − 24·116-s − 4·121-s + 32·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4-s + 3/4·16-s + 4.58·19-s − 2·25-s + 2.22·29-s − 2.87·31-s − 4.99·41-s + 16/7·49-s − 2.08·59-s − 1.02·61-s − 1/2·64-s − 0.949·71-s − 4.58·76-s + 1.80·79-s − 1.69·89-s + 2·100-s + 2.78·101-s + 3.44·109-s − 2.22·116-s − 0.363·121-s + 2.87·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.102540555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.102540555\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 142 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 94 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 814 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 + 20 T^{2} + 1558 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_4$ | \( ( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 7222 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 4 T^{2} - 698 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_4$ | \( ( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_4$ | \( ( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 8638 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 13558 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 8 T + 174 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 280 T^{2} + 36798 T^{4} - 280 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.07809524306551318955255060465, −6.91033823705214349534164527448, −6.72184024052943561211342506970, −6.07108019802061408245793658620, −5.92384333087476199343330538974, −5.90130607404530129969061503469, −5.76403880007138455769491397093, −5.29936802720990945089762401949, −5.10950365093518285800547889106, −4.98890474165195066997866544501, −4.89888016355259695651438855418, −4.56119144426292483518143247032, −4.29769733559596739031261590123, −3.77371278341399922541081479755, −3.67555559451482835302121707747, −3.36298921153972273060258796808, −3.35219085642063347919301023193, −3.08162627197731702120737937252, −2.84194299343676217612214822296, −2.14464046230874063237771012138, −1.96755688986959286829733024527, −1.57933089937505286183862185698, −1.24262176852662423923256635109, −0.915287690756451568720356901436, −0.26201167215684390288338673534,
0.26201167215684390288338673534, 0.915287690756451568720356901436, 1.24262176852662423923256635109, 1.57933089937505286183862185698, 1.96755688986959286829733024527, 2.14464046230874063237771012138, 2.84194299343676217612214822296, 3.08162627197731702120737937252, 3.35219085642063347919301023193, 3.36298921153972273060258796808, 3.67555559451482835302121707747, 3.77371278341399922541081479755, 4.29769733559596739031261590123, 4.56119144426292483518143247032, 4.89888016355259695651438855418, 4.98890474165195066997866544501, 5.10950365093518285800547889106, 5.29936802720990945089762401949, 5.76403880007138455769491397093, 5.90130607404530129969061503469, 5.92384333087476199343330538974, 6.07108019802061408245793658620, 6.72184024052943561211342506970, 6.91033823705214349534164527448, 7.07809524306551318955255060465
