| L(s) = 1 | − 2·3-s + 3·9-s − 6·13-s − 4·17-s − 8·23-s − 6·25-s − 4·27-s + 4·29-s + 12·39-s + 16·43-s − 49-s + 8·51-s + 12·53-s − 20·61-s + 16·69-s + 12·75-s − 24·79-s + 5·81-s − 8·87-s + 28·101-s − 16·103-s − 16·107-s + 28·113-s − 18·117-s + 22·121-s + 127-s − 32·129-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 1.66·13-s − 0.970·17-s − 1.66·23-s − 6/5·25-s − 0.769·27-s + 0.742·29-s + 1.92·39-s + 2.43·43-s − 1/7·49-s + 1.12·51-s + 1.64·53-s − 2.56·61-s + 1.92·69-s + 1.38·75-s − 2.70·79-s + 5/9·81-s − 0.857·87-s + 2.78·101-s − 1.57·103-s − 1.54·107-s + 2.63·113-s − 1.66·117-s + 2·121-s + 0.0887·127-s − 2.81·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3991660579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3991660579\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−8.592648872227311680537321122054, −7.87197703827494062877431580256, −7.86704772750699720547427361350, −7.42341071967536358943451076175, −7.06951750335072134300135227031, −6.76674053423287759388225383093, −6.22737762950374223145311289812, −5.98872454516788495442732501366, −5.53879240361378640278284856046, −5.48477662326818292399031119671, −4.63809081252335917557250494370, −4.44897987779151545281859609022, −4.30040158815505430621987789687, −3.79749672652095746115215417227, −3.04052540637661307078577740902, −2.62953165904748964823325901800, −2.03895980514155607256953307172, −1.84329804262241222572718503483, −0.913217991921513170717520337747, −0.22735640442195413357593259154,
0.22735640442195413357593259154, 0.913217991921513170717520337747, 1.84329804262241222572718503483, 2.03895980514155607256953307172, 2.62953165904748964823325901800, 3.04052540637661307078577740902, 3.79749672652095746115215417227, 4.30040158815505430621987789687, 4.44897987779151545281859609022, 4.63809081252335917557250494370, 5.48477662326818292399031119671, 5.53879240361378640278284856046, 5.98872454516788495442732501366, 6.22737762950374223145311289812, 6.76674053423287759388225383093, 7.06951750335072134300135227031, 7.42341071967536358943451076175, 7.86704772750699720547427361350, 7.87197703827494062877431580256, 8.592648872227311680537321122054
