| L(s) = 1 | + 3-s + 4-s + 5-s − 2·9-s + 3·11-s + 12-s + 13-s + 15-s + 16-s + 9·17-s + 20-s + 3·23-s − 4·25-s − 5·27-s − 8·31-s + 3·33-s − 2·36-s + 39-s + 3·44-s − 2·45-s + 48-s + 5·49-s + 9·51-s + 52-s − 9·53-s + 3·55-s + 60-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 0.447·5-s − 2/3·9-s + 0.904·11-s + 0.288·12-s + 0.277·13-s + 0.258·15-s + 1/4·16-s + 2.18·17-s + 0.223·20-s + 0.625·23-s − 4/5·25-s − 0.962·27-s − 1.43·31-s + 0.522·33-s − 1/3·36-s + 0.160·39-s + 0.452·44-s − 0.298·45-s + 0.144·48-s + 5/7·49-s + 1.26·51-s + 0.138·52-s − 1.23·53-s + 0.404·55-s + 0.129·60-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95220 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95220 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.375207731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.375207731\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 23 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 103 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 127 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
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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
−9.529375457009182882174230846368, −9.113516300238997131227353003419, −8.760906663714823780087879574240, −7.998342354022489993718252804831, −7.68377767501668301558490090118, −7.25765032168443191528709969150, −6.43569261671296485551821507349, −6.07691036489486510116449614378, −5.49038903335941488101221243595, −5.10988196301310675265866165587, −3.94851583576643495444488140638, −3.51618411085907967645385587580, −2.97292134790476139815548382321, −2.07489671932556335438282270884, −1.27323131034736585651463752327,
1.27323131034736585651463752327, 2.07489671932556335438282270884, 2.97292134790476139815548382321, 3.51618411085907967645385587580, 3.94851583576643495444488140638, 5.10988196301310675265866165587, 5.49038903335941488101221243595, 6.07691036489486510116449614378, 6.43569261671296485551821507349, 7.25765032168443191528709969150, 7.68377767501668301558490090118, 7.998342354022489993718252804831, 8.760906663714823780087879574240, 9.113516300238997131227353003419, 9.529375457009182882174230846368
