| L(s) = 1 | + 2·2-s − 2·3-s + 4-s − 2·5-s − 4·6-s − 9-s − 4·10-s + 4·11-s − 2·12-s + 4·13-s + 4·15-s + 16-s − 4·17-s − 2·18-s − 2·20-s + 8·22-s − 2·23-s + 3·25-s + 8·26-s + 6·27-s − 2·29-s + 8·30-s + 12·31-s − 2·32-s − 8·33-s − 8·34-s − 36-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s − 1.63·6-s − 1/3·9-s − 1.26·10-s + 1.20·11-s − 0.577·12-s + 1.10·13-s + 1.03·15-s + 1/4·16-s − 0.970·17-s − 0.471·18-s − 0.447·20-s + 1.70·22-s − 0.417·23-s + 3/5·25-s + 1.56·26-s + 1.15·27-s − 0.371·29-s + 1.46·30-s + 2.15·31-s − 0.353·32-s − 1.39·33-s − 1.37·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.486235741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.486235741\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 45 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 10 T + 99 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 109 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 22 T + 253 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 24 T + 294 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 155 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−12.29676671814416835404592247026, −11.93897411667216288364355734830, −11.30515886040948596263688304304, −11.25636786429156791956982373293, −10.88633594007959368871955684454, −10.16931436851515142485472574031, −9.353740640257808427434771484679, −8.998931123466527088577383332889, −8.225693104254845193097469473759, −8.037320149620824264731133241529, −6.99698434311428800477374270550, −6.61232254239300780776978889431, −5.91387810494493814029858157662, −5.87255390820298938816884145913, −4.83729408210650840005676871633, −4.61104680320551662806530745336, −3.85782578185694231530967713353, −3.64049755444429123970037675569, −2.49591203604440861640967031540, −0.901719564459636635872724397987,
0.901719564459636635872724397987, 2.49591203604440861640967031540, 3.64049755444429123970037675569, 3.85782578185694231530967713353, 4.61104680320551662806530745336, 4.83729408210650840005676871633, 5.87255390820298938816884145913, 5.91387810494493814029858157662, 6.61232254239300780776978889431, 6.99698434311428800477374270550, 8.037320149620824264731133241529, 8.225693104254845193097469473759, 8.998931123466527088577383332889, 9.353740640257808427434771484679, 10.16931436851515142485472574031, 10.88633594007959368871955684454, 11.25636786429156791956982373293, 11.30515886040948596263688304304, 11.93897411667216288364355734830, 12.29676671814416835404592247026
