L(s) = 1 | + 8·7-s + 12·13-s − 8·25-s + 16·31-s + 12·37-s + 32·49-s + 16·67-s + 4·73-s + 96·91-s − 12·97-s − 8·103-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s − 64·175-s + 179-s + 181-s + 191-s + ⋯ |
L(s) = 1 | + 3.02·7-s + 3.32·13-s − 8/5·25-s + 2.87·31-s + 1.97·37-s + 32/7·49-s + 1.95·67-s + 0.468·73-s + 10.0·91-s − 1.21·97-s − 0.788·103-s + 2.54·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.53·169-s + 0.0760·173-s − 4.83·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(16.93263487\) |
\(L(\frac12)\) |
\(\approx\) |
\(16.93263487\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
good | 7 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 254 T^{4} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )( 1 + 56 T^{2} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 13294 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.34680554952465658413442617093, −5.85198347196162323719173142915, −5.75177833443346919381436300328, −5.62689610577922330789261817920, −5.50318288069425679045160222140, −5.26535313381007075782316190169, −4.94807482235245313444545334742, −4.68761320039714464333298125407, −4.48130764422735073818076733285, −4.33926174148688579169858580310, −4.21108431978132632214950986085, −4.02125785735336972221812949866, −3.91603426359309901457247285952, −3.32291430617390839319751358082, −3.21124979347463454741113106613, −3.19047766065095633938807696706, −2.70773815557248207783827620141, −2.34968966430422350903785234190, −2.04822708459666890835639199135, −1.94346725121839294998128951894, −1.68628533048522070505094576715, −1.35614319664647930855672401145, −1.04451784357876665915434651131, −0.78614056538542053518017382597, −0.70951641571912018364796833547,
0.70951641571912018364796833547, 0.78614056538542053518017382597, 1.04451784357876665915434651131, 1.35614319664647930855672401145, 1.68628533048522070505094576715, 1.94346725121839294998128951894, 2.04822708459666890835639199135, 2.34968966430422350903785234190, 2.70773815557248207783827620141, 3.19047766065095633938807696706, 3.21124979347463454741113106613, 3.32291430617390839319751358082, 3.91603426359309901457247285952, 4.02125785735336972221812949866, 4.21108431978132632214950986085, 4.33926174148688579169858580310, 4.48130764422735073818076733285, 4.68761320039714464333298125407, 4.94807482235245313444545334742, 5.26535313381007075782316190169, 5.50318288069425679045160222140, 5.62689610577922330789261817920, 5.75177833443346919381436300328, 5.85198347196162323719173142915, 6.34680554952465658413442617093