L(s) = 1 | + 2·5-s − 3·9-s − 6·13-s + 2·17-s − 25-s + 10·29-s + 2·37-s + 10·41-s − 6·45-s − 7·49-s − 14·53-s + 10·61-s − 12·65-s − 6·73-s + 9·81-s + 4·85-s + 10·89-s + 18·97-s + 2·101-s − 6·109-s − 14·113-s + 18·117-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 9-s − 1.66·13-s + 0.485·17-s − 1/5·25-s + 1.85·29-s + 0.328·37-s + 1.56·41-s − 0.894·45-s − 49-s − 1.92·53-s + 1.28·61-s − 1.48·65-s − 0.702·73-s + 81-s + 0.433·85-s + 1.05·89-s + 1.82·97-s + 0.199·101-s − 0.574·109-s − 1.31·113-s + 1.66·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9270373386\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9270373386\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.59250078389593402269438368141, −14.09304813859309984769942580998, −12.72487277908244699547747964129, −11.66649219374340327203675385621, −10.23421229877843747435058592930, −9.317356556158666156286958675354, −7.86574055998535746161422058704, −6.27282080646876620831649653268, −5.01452297596172644651029655270, −2.63895104553179739845027310173,
2.63895104553179739845027310173, 5.01452297596172644651029655270, 6.27282080646876620831649653268, 7.86574055998535746161422058704, 9.317356556158666156286958675354, 10.23421229877843747435058592930, 11.66649219374340327203675385621, 12.72487277908244699547747964129, 14.09304813859309984769942580998, 14.59250078389593402269438368141