L(s) = 1 | + 2·11-s − 13-s − 5·17-s + 19-s − 23-s − 5·25-s + 3·29-s + 4·31-s + 2·37-s − 8·41-s + 8·43-s + 8·47-s − 9·53-s − 59-s − 14·61-s − 13·67-s + 10·71-s − 9·73-s + 10·79-s − 10·83-s − 12·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | + 0.603·11-s − 0.277·13-s − 1.21·17-s + 0.229·19-s − 0.208·23-s − 25-s + 0.557·29-s + 0.718·31-s + 0.328·37-s − 1.24·41-s + 1.21·43-s + 1.16·47-s − 1.23·53-s − 0.130·59-s − 1.79·61-s − 1.58·67-s + 1.18·71-s − 1.05·73-s + 1.12·79-s − 1.09·83-s − 1.27·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.470677867\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.470677867\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.65947518132240, −12.92913872537378, −12.41317100736612, −12.09064315328145, −11.44238292445691, −11.21818422599774, −10.48366849522540, −10.14842344461582, −9.464687499114007, −9.159180529475385, −8.623646105605109, −8.069566126230843, −7.553334239049150, −7.030601675693221, −6.478331259186494, −6.036595926593245, −5.550081767216520, −4.647108173951712, −4.456675257532921, −3.850819845371787, −3.095633854936846, −2.582707695839856, −1.872784625214306, −1.295722475370169, −0.3646310267057184,
0.3646310267057184, 1.295722475370169, 1.872784625214306, 2.582707695839856, 3.095633854936846, 3.850819845371787, 4.456675257532921, 4.647108173951712, 5.550081767216520, 6.036595926593245, 6.478331259186494, 7.030601675693221, 7.553334239049150, 8.069566126230843, 8.623646105605109, 9.159180529475385, 9.464687499114007, 10.14842344461582, 10.48366849522540, 11.21818422599774, 11.44238292445691, 12.09064315328145, 12.41317100736612, 12.92913872537378, 13.65947518132240