| L(s) = 1 | + 2·3-s + 2·5-s + 3·9-s − 4·11-s + 4·15-s − 25-s + 4·27-s + 8·31-s − 8·33-s + 8·37-s + 6·45-s + 8·47-s − 2·49-s − 8·55-s − 8·59-s + 16·67-s − 16·71-s − 2·75-s + 5·81-s − 12·89-s + 16·93-s + 24·97-s − 12·99-s + 16·111-s + 16·113-s + 5·121-s − 12·125-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 9-s − 1.20·11-s + 1.03·15-s − 1/5·25-s + 0.769·27-s + 1.43·31-s − 1.39·33-s + 1.31·37-s + 0.894·45-s + 1.16·47-s − 2/7·49-s − 1.07·55-s − 1.04·59-s + 1.95·67-s − 1.89·71-s − 0.230·75-s + 5/9·81-s − 1.27·89-s + 1.65·93-s + 2.43·97-s − 1.20·99-s + 1.51·111-s + 1.50·113-s + 5/11·121-s − 1.07·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.972252962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.972252962\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
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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
−7.889283016124604108044912797824, −7.53523051201443360725153436617, −7.12522845297380770033083124507, −6.48196107872506837099957151060, −6.15196177880609936761154956654, −5.70579025603662769757837522570, −5.20692549658435572742795424473, −4.67553342080002641787364791728, −4.29884749351892202431719319297, −3.71118253841990386588279525987, −2.97148807225958338569498293311, −2.74629713889120504754939574949, −2.21824829757854020211257784117, −1.67856463829489378871226519803, −0.77122585709305171950094430743,
0.77122585709305171950094430743, 1.67856463829489378871226519803, 2.21824829757854020211257784117, 2.74629713889120504754939574949, 2.97148807225958338569498293311, 3.71118253841990386588279525987, 4.29884749351892202431719319297, 4.67553342080002641787364791728, 5.20692549658435572742795424473, 5.70579025603662769757837522570, 6.15196177880609936761154956654, 6.48196107872506837099957151060, 7.12522845297380770033083124507, 7.53523051201443360725153436617, 7.889283016124604108044912797824
