| L(s) = 1 | + 2-s + 4-s − 3·5-s − 2·7-s + 8-s − 3·10-s + 6·11-s − 2·14-s + 16-s − 3·20-s + 6·22-s + 4·25-s − 2·28-s + 32-s + 6·35-s − 3·40-s − 20·43-s + 6·44-s − 11·49-s + 4·50-s − 18·53-s − 18·55-s − 2·56-s − 24·59-s + 16·61-s + 64-s + 28·67-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.755·7-s + 0.353·8-s − 0.948·10-s + 1.80·11-s − 0.534·14-s + 1/4·16-s − 0.670·20-s + 1.27·22-s + 4/5·25-s − 0.377·28-s + 0.176·32-s + 1.01·35-s − 0.474·40-s − 3.04·43-s + 0.904·44-s − 1.57·49-s + 0.565·50-s − 2.47·53-s − 2.42·55-s − 0.267·56-s − 3.12·59-s + 2.04·61-s + 1/8·64-s + 3.42·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 583200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 583200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−8.175508045180459617652224529359, −7.82723843094092867969941089213, −7.07832589673054274134741987475, −6.71508725988629390136097161445, −6.50809047734617527891647867661, −6.10384069202259277792945056161, −5.26189135061292063008867608145, −4.77709333112371596512170551343, −4.38306493815359613482087430509, −3.72351464227480953738971596775, −3.38549178369413487864105189864, −3.17052507023677297081073811536, −2.00274833407885544645947603057, −1.28826140651909171078200157979, 0,
1.28826140651909171078200157979, 2.00274833407885544645947603057, 3.17052507023677297081073811536, 3.38549178369413487864105189864, 3.72351464227480953738971596775, 4.38306493815359613482087430509, 4.77709333112371596512170551343, 5.26189135061292063008867608145, 6.10384069202259277792945056161, 6.50809047734617527891647867661, 6.71508725988629390136097161445, 7.07832589673054274134741987475, 7.82723843094092867969941089213, 8.175508045180459617652224529359
