| L(s) = 1 | + 9-s − 2·13-s − 4·17-s − 4·29-s + 4·41-s + 2·49-s − 2·53-s − 2·61-s − 2·73-s + 2·97-s + 4·101-s − 2·109-s + 4·113-s − 2·117-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + 167-s + 169-s + 173-s + ⋯ |
| L(s) = 1 | + 9-s − 2·13-s − 4·17-s − 4·29-s + 4·41-s + 2·49-s − 2·53-s − 2·61-s − 2·73-s + 2·97-s + 4·101-s − 2·109-s + 4·113-s − 2·117-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + 167-s + 169-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5720994699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5720994699\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 19 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 31 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 53 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 59 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 61 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 67 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 73 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.16307091778900805176193829291, −5.97569036803006198942424979041, −5.87338519582092453275473412395, −5.57281628010657758576924499425, −5.40817734288491161454754442675, −5.05116724938903643651891765014, −4.92229870832455759520692965289, −4.68061046561209791955590972037, −4.51178636250351241981734582257, −4.32464146740364387940538477509, −4.27921128706487759394645745623, −4.02583803413308960917667996309, −3.98671781572133616016195783297, −3.35866264594108745200205293175, −3.33921066949657398906188169899, −3.16281552439781541146264152255, −2.65755440860477085356212857283, −2.35624710858523781555697740030, −2.34021355019288633650008374236, −2.10026071645969598454327607370, −2.06816803820930480604423905075, −1.52140698720427526983637471325, −1.49857196349192244242743947173, −0.74316149484627271561463929145, −0.30685085800694714946030010550,
0.30685085800694714946030010550, 0.74316149484627271561463929145, 1.49857196349192244242743947173, 1.52140698720427526983637471325, 2.06816803820930480604423905075, 2.10026071645969598454327607370, 2.34021355019288633650008374236, 2.35624710858523781555697740030, 2.65755440860477085356212857283, 3.16281552439781541146264152255, 3.33921066949657398906188169899, 3.35866264594108745200205293175, 3.98671781572133616016195783297, 4.02583803413308960917667996309, 4.27921128706487759394645745623, 4.32464146740364387940538477509, 4.51178636250351241981734582257, 4.68061046561209791955590972037, 4.92229870832455759520692965289, 5.05116724938903643651891765014, 5.40817734288491161454754442675, 5.57281628010657758576924499425, 5.87338519582092453275473412395, 5.97569036803006198942424979041, 6.16307091778900805176193829291
