| L(s) = 1 | − 6·9-s + 4·11-s − 4·16-s − 6·23-s − 10·25-s − 20·31-s + 14·37-s − 22·47-s − 5·49-s + 20·53-s − 20·67-s + 16·71-s + 27·81-s − 16·89-s − 4·97-s − 24·99-s − 8·103-s − 8·113-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 24·144-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 2·9-s + 1.20·11-s − 16-s − 1.25·23-s − 2·25-s − 3.59·31-s + 2.30·37-s − 3.20·47-s − 5/7·49-s + 2.74·53-s − 2.44·67-s + 1.89·71-s + 3·81-s − 1.69·89-s − 0.406·97-s − 2.41·99-s − 0.788·103-s − 0.752·113-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4695889 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4695889 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 197 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−6.92472041340395206972505718477, −6.32085609198500063533870245283, −6.17591940844279569664203110179, −5.76522191873394385518747749641, −5.38321485974197223099421779889, −5.01602374926633435775396444233, −4.15785519193072071590213469433, −3.99653125016172072692387582493, −3.57456424301634246955419853703, −3.03314726794163826214204324634, −2.33277116067346011831864956690, −2.04363179109126975355030360625, −1.40921778729921431995637127835, 0, 0,
1.40921778729921431995637127835, 2.04363179109126975355030360625, 2.33277116067346011831864956690, 3.03314726794163826214204324634, 3.57456424301634246955419853703, 3.99653125016172072692387582493, 4.15785519193072071590213469433, 5.01602374926633435775396444233, 5.38321485974197223099421779889, 5.76522191873394385518747749641, 6.17591940844279569664203110179, 6.32085609198500063533870245283, 6.92472041340395206972505718477
