| 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
| \(L(1)\) |
\(\approx\) |
\(0.4543540219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4543540219\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 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$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 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$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 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
−8.628627084335157344896895932287, −8.162509941803446296617357223222, −7.65763656260689066974320727826, −7.17917249986642937603660017926, −6.45210955534301586661244338933, −6.31650311880734045687516863274, −5.87925810986092187606144516601, −5.37233624559303552365936514868, −4.91388785325285824274569090174, −4.74412577109589626998638969705, −3.78322327485312266891934259156, −3.25248838121025487875312938747, −2.35759312408797877080448268486, −1.36059232511906606923189065222, −0.46007285431386247558305501616,
0.46007285431386247558305501616, 1.36059232511906606923189065222, 2.35759312408797877080448268486, 3.25248838121025487875312938747, 3.78322327485312266891934259156, 4.74412577109589626998638969705, 4.91388785325285824274569090174, 5.37233624559303552365936514868, 5.87925810986092187606144516601, 6.31650311880734045687516863274, 6.45210955534301586661244338933, 7.17917249986642937603660017926, 7.65763656260689066974320727826, 8.162509941803446296617357223222, 8.628627084335157344896895932287
