| L(s) = 1 | − 3-s − 8·7-s + 9-s − 8·13-s + 8·21-s + 2·25-s − 27-s + 4·31-s + 8·39-s + 8·43-s + 34·49-s + 12·61-s − 8·63-s − 8·67-s + 4·73-s − 2·75-s + 20·79-s + 81-s + 64·91-s − 4·93-s + 20·97-s − 12·103-s − 8·117-s + 18·121-s + 127-s − 8·129-s + 131-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 3.02·7-s + 1/3·9-s − 2.21·13-s + 1.74·21-s + 2/5·25-s − 0.192·27-s + 0.718·31-s + 1.28·39-s + 1.21·43-s + 34/7·49-s + 1.53·61-s − 1.00·63-s − 0.977·67-s + 0.468·73-s − 0.230·75-s + 2.25·79-s + 1/9·81-s + 6.70·91-s − 0.414·93-s + 2.03·97-s − 1.18·103-s − 0.739·117-s + 1.63·121-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−8.043159863683799570763951127685, −7.51532693869291336446784743986, −7.07540537849845782365416449733, −6.84200140138577701878048003613, −6.32277177115757490874269891405, −6.06181105053951209160133781140, −5.46738204160296595993457540347, −4.92004909346387105610574681262, −4.44155374982899169682272065246, −3.67048279266388787159699737632, −3.33387171734254432326114515960, −2.56189884412683094767219908503, −2.39042856756819379747711924801, −0.74946925825542088720186972596, 0,
0.74946925825542088720186972596, 2.39042856756819379747711924801, 2.56189884412683094767219908503, 3.33387171734254432326114515960, 3.67048279266388787159699737632, 4.44155374982899169682272065246, 4.92004909346387105610574681262, 5.46738204160296595993457540347, 6.06181105053951209160133781140, 6.32277177115757490874269891405, 6.84200140138577701878048003613, 7.07540537849845782365416449733, 7.51532693869291336446784743986, 8.043159863683799570763951127685
