| L(s) = 1 | + 2·3-s − 2·4-s + 7-s − 3·9-s − 4·12-s + 4·16-s + 4·19-s + 2·21-s + 25-s − 14·27-s − 2·28-s + 6·29-s − 8·31-s + 6·36-s + 4·37-s + 18·47-s + 8·48-s + 49-s + 24·53-s + 8·57-s − 3·63-s − 8·64-s + 2·75-s − 8·76-s − 4·81-s + 24·83-s − 4·84-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s + 0.377·7-s − 9-s − 1.15·12-s + 16-s + 0.917·19-s + 0.436·21-s + 1/5·25-s − 2.69·27-s − 0.377·28-s + 1.11·29-s − 1.43·31-s + 36-s + 0.657·37-s + 2.62·47-s + 1.15·48-s + 1/7·49-s + 3.29·53-s + 1.05·57-s − 0.377·63-s − 64-s + 0.230·75-s − 0.917·76-s − 4/9·81-s + 2.63·83-s − 0.436·84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.680716502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680716502\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−9.223798828824354541962014191697, −8.703677986065848624605724183186, −8.679498853252053444462430241617, −7.980947296320422939728903060714, −7.59679103544400835112633947355, −7.18840908028866321022100981317, −6.23576189288857341841080164821, −5.51580216499301883688612135353, −5.47907564040983295563221596641, −4.66901616031576415432379359350, −3.81894678634232741164227871759, −3.61689003045514298236484607491, −2.72282955438852544862637314814, −2.28641578958231941287903879491, −0.863841609060871512635709852891,
0.863841609060871512635709852891, 2.28641578958231941287903879491, 2.72282955438852544862637314814, 3.61689003045514298236484607491, 3.81894678634232741164227871759, 4.66901616031576415432379359350, 5.47907564040983295563221596641, 5.51580216499301883688612135353, 6.23576189288857341841080164821, 7.18840908028866321022100981317, 7.59679103544400835112633947355, 7.980947296320422939728903060714, 8.679498853252053444462430241617, 8.703677986065848624605724183186, 9.223798828824354541962014191697
