L(s) = 1 | + 3-s − 2·5-s + 9-s − 6·13-s − 2·15-s − 17-s + 8·23-s − 25-s + 27-s + 6·29-s − 8·31-s − 10·37-s − 6·39-s + 6·41-s + 12·43-s − 2·45-s − 51-s + 10·53-s + 8·59-s + 6·61-s + 12·65-s + 12·67-s + 8·69-s + 6·73-s − 75-s + 8·79-s + 81-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.66·13-s − 0.516·15-s − 0.242·17-s + 1.66·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 1.64·37-s − 0.960·39-s + 0.937·41-s + 1.82·43-s − 0.298·45-s − 0.140·51-s + 1.37·53-s + 1.04·59-s + 0.768·61-s + 1.48·65-s + 1.46·67-s + 0.963·69-s + 0.702·73-s − 0.115·75-s + 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.074986690\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.074986690\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.02037265754001, −12.85250248353671, −12.41256073776550, −11.84542174775340, −11.50599041853700, −10.78487153304435, −10.55529526242800, −9.804025891454910, −9.447702244294303, −8.871387265251011, −8.531015187442713, −7.899175355871409, −7.398768249125269, −7.031167571965050, −6.807613406515492, −5.737340630905546, −5.270350576437094, −4.763904587223193, −4.206065192230252, −3.690496682515542, −3.151441561935036, −2.398860640664225, −2.194739188063611, −1.067616928800154, −0.4504609824128618,
0.4504609824128618, 1.067616928800154, 2.194739188063611, 2.398860640664225, 3.151441561935036, 3.690496682515542, 4.206065192230252, 4.763904587223193, 5.270350576437094, 5.737340630905546, 6.807613406515492, 7.031167571965050, 7.398768249125269, 7.899175355871409, 8.531015187442713, 8.871387265251011, 9.447702244294303, 9.804025891454910, 10.55529526242800, 10.78487153304435, 11.50599041853700, 11.84542174775340, 12.41256073776550, 12.85250248353671, 13.02037265754001