| L(s) = 1 | − 92·25-s − 4·49-s − 200·73-s − 760·97-s + 284·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 676·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 3.67·25-s − 0.0816·49-s − 2.73·73-s − 7.83·97-s + 2.34·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 4·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1416819133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1416819133\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2}( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 29 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{2}( 1 + 50 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 478 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 53 | $C_2$ | \( ( 1 - 94 T + p^{2} T^{2} )^{2}( 1 + 94 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6862 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 9118 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 4178 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 190 T + p^{2} T^{2} )^{4} \) |
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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
−7.58373137262036577090294129325, −7.55740184898913870140827545297, −6.95503173225092534902817472698, −6.72193461930529100778102954511, −6.59380575600429554550151008457, −6.46563584981234014544888712075, −5.86659425715633789159343972901, −5.76595245986668593765952958112, −5.61935304011932521478655205358, −5.41155104262349139758242554523, −5.26426332892031628858842588264, −4.64552878120471088945286970527, −4.30075202164223307095956473120, −4.28821947232849263392740163281, −3.94494754555310115294741397744, −3.90946028406109882204234610092, −3.25578682460969150495381607858, −2.99085887228669528185934716588, −2.89479902782466856688798232670, −2.33730674684002617368207483186, −1.98975151081958697349444020617, −1.61456567397340796589146789630, −1.49909587933444669779976361458, −0.73714163175307104414493199133, −0.07547772934728026695330522549,
0.07547772934728026695330522549, 0.73714163175307104414493199133, 1.49909587933444669779976361458, 1.61456567397340796589146789630, 1.98975151081958697349444020617, 2.33730674684002617368207483186, 2.89479902782466856688798232670, 2.99085887228669528185934716588, 3.25578682460969150495381607858, 3.90946028406109882204234610092, 3.94494754555310115294741397744, 4.28821947232849263392740163281, 4.30075202164223307095956473120, 4.64552878120471088945286970527, 5.26426332892031628858842588264, 5.41155104262349139758242554523, 5.61935304011932521478655205358, 5.76595245986668593765952958112, 5.86659425715633789159343972901, 6.46563584981234014544888712075, 6.59380575600429554550151008457, 6.72193461930529100778102954511, 6.95503173225092534902817472698, 7.55740184898913870140827545297, 7.58373137262036577090294129325
