| L(s) = 1 | + 4·3-s + 7·9-s + 4·11-s + 4·13-s + 2·17-s + 4·19-s − 8·23-s − 5·25-s + 4·27-s + 16·33-s + 16·39-s + 8·47-s − 5·49-s + 8·51-s + 16·57-s + 8·67-s − 32·69-s + 12·73-s − 20·75-s − 8·79-s − 8·81-s + 14·89-s + 28·99-s − 16·103-s + 28·117-s + 5·121-s + 127-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 7/3·9-s + 1.20·11-s + 1.10·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s − 25-s + 0.769·27-s + 2.78·33-s + 2.56·39-s + 1.16·47-s − 5/7·49-s + 1.12·51-s + 2.11·57-s + 0.977·67-s − 3.85·69-s + 1.40·73-s − 2.30·75-s − 0.900·79-s − 8/9·81-s + 1.48·89-s + 2.81·99-s − 1.57·103-s + 2.58·117-s + 5/11·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.775128322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.775128322\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
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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
−8.492787059234485421774562671494, −8.289284972448356246648997464919, −7.976428214382001392250982113893, −7.54870133691104598222479867437, −6.97781732298805289348156183618, −6.41586235922114715157533893818, −5.86967337820204070883730180230, −5.48270529046645535483049251847, −4.47016105343047367538779745130, −3.94950946519446386652926346287, −3.55025539406315492226642404721, −3.31942765475398743069821339984, −2.44872044284472551883744097998, −1.95996517575261991484716883973, −1.20587031736466145751119695485,
1.20587031736466145751119695485, 1.95996517575261991484716883973, 2.44872044284472551883744097998, 3.31942765475398743069821339984, 3.55025539406315492226642404721, 3.94950946519446386652926346287, 4.47016105343047367538779745130, 5.48270529046645535483049251847, 5.86967337820204070883730180230, 6.41586235922114715157533893818, 6.97781732298805289348156183618, 7.54870133691104598222479867437, 7.976428214382001392250982113893, 8.289284972448356246648997464919, 8.492787059234485421774562671494
