| L(s) = 1 | + 2·2-s − 4-s − 8·8-s − 7·16-s − 12·17-s + 8·25-s − 16·29-s − 16·31-s + 14·32-s − 24·34-s − 20·37-s + 6·49-s + 16·50-s − 32·58-s − 32·62-s + 35·64-s − 8·67-s + 12·68-s − 40·74-s − 8·83-s + 20·97-s + 12·98-s − 8·100-s + 16·101-s − 8·103-s + 16·107-s + 16·116-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s − 2.82·8-s − 7/4·16-s − 2.91·17-s + 8/5·25-s − 2.97·29-s − 2.87·31-s + 2.47·32-s − 4.11·34-s − 3.28·37-s + 6/7·49-s + 2.26·50-s − 4.20·58-s − 4.06·62-s + 35/8·64-s − 0.977·67-s + 1.45·68-s − 4.64·74-s − 0.878·83-s + 2.03·97-s + 1.21·98-s − 4/5·100-s + 1.59·101-s − 0.788·103-s + 1.54·107-s + 1.48·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3198280675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3198280675\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 104 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 72 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 96 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 160 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−10.28386302674145506190150135339, −9.270767444191115457050762504746, −9.087382126779537228013400247292, −8.937983384080349493656413036513, −8.821530432246395056577364210723, −8.121550675525604134772370416068, −7.33527517311975019436416413494, −6.97378168104924839469547757038, −6.78313108307094394755798447436, −5.89082107297916174387086381749, −5.76871584237633525626608958480, −5.20396691560226240479717611412, −4.89830335170751826937406022994, −4.41770025287563331225164272071, −3.97494312672525015030931134646, −3.44197402475290713933458150761, −3.30344458171355219951648282808, −2.20554667488558793374068410950, −1.87541829871856476581948085132, −0.19026246820860583551012464840,
0.19026246820860583551012464840, 1.87541829871856476581948085132, 2.20554667488558793374068410950, 3.30344458171355219951648282808, 3.44197402475290713933458150761, 3.97494312672525015030931134646, 4.41770025287563331225164272071, 4.89830335170751826937406022994, 5.20396691560226240479717611412, 5.76871584237633525626608958480, 5.89082107297916174387086381749, 6.78313108307094394755798447436, 6.97378168104924839469547757038, 7.33527517311975019436416413494, 8.121550675525604134772370416068, 8.821530432246395056577364210723, 8.937983384080349493656413036513, 9.087382126779537228013400247292, 9.270767444191115457050762504746, 10.28386302674145506190150135339
