L(s) = 1 | − 2·9-s + 8·13-s + 12·17-s − 10·25-s + 12·29-s − 4·37-s + 12·41-s + 49-s − 12·53-s − 16·61-s + 4·73-s − 5·81-s − 12·89-s − 20·97-s − 4·109-s + 12·113-s − 16·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 163-s + ⋯ |
L(s) = 1 | − 2/3·9-s + 2.21·13-s + 2.91·17-s − 2·25-s + 2.22·29-s − 0.657·37-s + 1.87·41-s + 1/7·49-s − 1.64·53-s − 2.04·61-s + 0.468·73-s − 5/9·81-s − 1.27·89-s − 2.03·97-s − 0.383·109-s + 1.12·113-s − 1.47·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.94·153-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.969688554\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.969688554\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−9.195409247501138715663586603039, −8.328602349022309226048251750698, −8.222985047096640302551257734717, −7.87021772021814993834479658639, −7.29200576171944643329126204500, −6.47781377385923123584313505983, −6.04992362572003306544038558924, −5.73625972355331222443538857924, −5.36663802373794330843690451326, −4.39126934619586232082133319537, −3.93522119575008455702519349301, −3.09628357761679869111062402047, −3.07302685938302839530976846531, −1.64950502134773404742708641242, −1.00403210582709168291350506092,
1.00403210582709168291350506092, 1.64950502134773404742708641242, 3.07302685938302839530976846531, 3.09628357761679869111062402047, 3.93522119575008455702519349301, 4.39126934619586232082133319537, 5.36663802373794330843690451326, 5.73625972355331222443538857924, 6.04992362572003306544038558924, 6.47781377385923123584313505983, 7.29200576171944643329126204500, 7.87021772021814993834479658639, 8.222985047096640302551257734717, 8.328602349022309226048251750698, 9.195409247501138715663586603039