| L(s) = 1 | + 3-s − 4·5-s + 9-s − 4·13-s − 4·15-s + 2·17-s + 2·25-s + 27-s + 12·29-s − 16·31-s − 4·39-s − 12·41-s − 4·45-s − 14·49-s + 2·51-s − 8·59-s + 16·65-s + 2·75-s + 16·79-s + 81-s + 8·83-s − 8·85-s + 12·87-s − 16·93-s + 36·113-s − 4·117-s − 6·121-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 1/3·9-s − 1.10·13-s − 1.03·15-s + 0.485·17-s + 2/5·25-s + 0.192·27-s + 2.22·29-s − 2.87·31-s − 0.640·39-s − 1.87·41-s − 0.596·45-s − 2·49-s + 0.280·51-s − 1.04·59-s + 1.98·65-s + 0.230·75-s + 1.80·79-s + 1/9·81-s + 0.878·83-s − 0.867·85-s + 1.28·87-s − 1.65·93-s + 3.38·113-s − 0.369·117-s − 0.545·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 998784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 998784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7958503784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7958503784\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.173282349834764667604075089133, −7.66498871915988097333678887917, −7.26958585488936859387169650645, −7.18155599764795540835879645469, −6.42703115208668412831343638258, −6.03138344128444123566991871629, −5.04277040000068891801244440280, −5.01357758335939619070346679818, −4.42764839672826892298257893061, −3.80218561106952661254790284491, −3.39638272741083957218437360098, −3.14826068230388029805668446658, −2.23113257513110131986739736948, −1.62857614603859772213021712939, −0.39064935037816856965515086908,
0.39064935037816856965515086908, 1.62857614603859772213021712939, 2.23113257513110131986739736948, 3.14826068230388029805668446658, 3.39638272741083957218437360098, 3.80218561106952661254790284491, 4.42764839672826892298257893061, 5.01357758335939619070346679818, 5.04277040000068891801244440280, 6.03138344128444123566991871629, 6.42703115208668412831343638258, 7.18155599764795540835879645469, 7.26958585488936859387169650645, 7.66498871915988097333678887917, 8.173282349834764667604075089133
