| L(s) = 1 | + 2·3-s − 2·4-s − 3·9-s − 6·11-s − 4·12-s + 4·16-s + 6·17-s + 4·19-s + 25-s − 14·27-s − 12·33-s + 6·36-s − 24·41-s − 20·43-s + 12·44-s + 8·48-s + 49-s + 12·51-s + 8·57-s − 8·64-s − 8·67-s − 12·68-s + 4·73-s + 2·75-s − 8·76-s − 4·81-s + 24·83-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s − 9-s − 1.80·11-s − 1.15·12-s + 16-s + 1.45·17-s + 0.917·19-s + 1/5·25-s − 2.69·27-s − 2.08·33-s + 36-s − 3.74·41-s − 3.04·43-s + 1.80·44-s + 1.15·48-s + 1/7·49-s + 1.68·51-s + 1.05·57-s − 64-s − 0.977·67-s − 1.45·68-s + 0.468·73-s + 0.230·75-s − 0.917·76-s − 4/9·81-s + 2.63·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.581459961422691318510933413886, −8.679498853252053444462430241617, −8.611548044991135538391646913928, −7.980947296320422939728903060714, −7.937110667039146160138689071489, −7.21068288194212183905414731057, −6.34730906119503780706199760723, −5.47907564040983295563221596641, −5.25086715638915413911306599520, −4.93400384358317586060280833112, −3.61689003045514298236484607491, −3.24103653131434423009222437756, −2.94130668011304353115387148308, −1.79896277633481319941660230821, 0,
1.79896277633481319941660230821, 2.94130668011304353115387148308, 3.24103653131434423009222437756, 3.61689003045514298236484607491, 4.93400384358317586060280833112, 5.25086715638915413911306599520, 5.47907564040983295563221596641, 6.34730906119503780706199760723, 7.21068288194212183905414731057, 7.937110667039146160138689071489, 7.980947296320422939728903060714, 8.611548044991135538391646913928, 8.679498853252053444462430241617, 9.581459961422691318510933413886
