| L(s) = 1 | − 2·3-s + 2·5-s + 2·7-s + 3·9-s − 4·13-s − 4·15-s − 6·17-s − 2·19-s − 4·21-s − 4·25-s − 4·27-s − 2·29-s + 4·31-s + 4·35-s − 20·37-s + 8·39-s − 4·41-s − 4·43-s + 6·45-s − 6·47-s + 3·49-s + 12·51-s − 10·53-s + 4·57-s + 16·59-s − 4·61-s + 6·63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 0.755·7-s + 9-s − 1.10·13-s − 1.03·15-s − 1.45·17-s − 0.458·19-s − 0.872·21-s − 4/5·25-s − 0.769·27-s − 0.371·29-s + 0.718·31-s + 0.676·35-s − 3.28·37-s + 1.28·39-s − 0.624·41-s − 0.609·43-s + 0.894·45-s − 0.875·47-s + 3/7·49-s + 1.68·51-s − 1.37·53-s + 0.529·57-s + 2.08·59-s − 0.512·61-s + 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10188864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10188864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 2 T - 16 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 128 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 164 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 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
−8.392421729185139381000386733539, −8.298109307110716885772338473584, −7.49360831030062587068886618093, −7.38990802393262026154540378761, −6.74467953239146270778864976808, −6.72020247256648067811399667989, −6.11060465572420654094912371029, −5.96050108383078022144209583258, −5.17630396496452879661708451088, −5.16013402141746302475978985311, −4.78673596878036509175024432651, −4.46483135621756405828761423911, −3.70580254954032132318419206451, −3.51852931910054145811193539037, −2.48921515589753671598921771542, −2.25127277153603816838085760738, −1.68195776420004564847209221213, −1.38790290046209258917444340189, 0, 0,
1.38790290046209258917444340189, 1.68195776420004564847209221213, 2.25127277153603816838085760738, 2.48921515589753671598921771542, 3.51852931910054145811193539037, 3.70580254954032132318419206451, 4.46483135621756405828761423911, 4.78673596878036509175024432651, 5.16013402141746302475978985311, 5.17630396496452879661708451088, 5.96050108383078022144209583258, 6.11060465572420654094912371029, 6.72020247256648067811399667989, 6.74467953239146270778864976808, 7.38990802393262026154540378761, 7.49360831030062587068886618093, 8.298109307110716885772338473584, 8.392421729185139381000386733539
