L(s) = 1 | − 3-s − 2·5-s + 9-s − 11-s − 2·13-s + 2·15-s + 17-s − 19-s + 9·23-s − 25-s − 27-s + 9·29-s − 8·31-s + 33-s + 7·37-s + 2·39-s + 10·41-s + 9·43-s − 2·45-s − 9·47-s − 51-s + 6·53-s + 2·55-s + 57-s + 13·59-s − 4·61-s + 4·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.516·15-s + 0.242·17-s − 0.229·19-s + 1.87·23-s − 1/5·25-s − 0.192·27-s + 1.67·29-s − 1.43·31-s + 0.174·33-s + 1.15·37-s + 0.320·39-s + 1.56·41-s + 1.37·43-s − 0.298·45-s − 1.31·47-s − 0.140·51-s + 0.824·53-s + 0.269·55-s + 0.132·57-s + 1.69·59-s − 0.512·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.839343038\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.839343038\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−13.60684397945671, −13.05827815559303, −12.63544852750295, −12.36808896008204, −11.62815514193258, −11.34182378042626, −10.88436557636456, −10.42062094762185, −9.811340900223808, −9.277682855229049, −8.781973406613288, −8.149979733027516, −7.552609187442312, −7.347159066727911, −6.671018959377150, −6.170926643113722, −5.423885879718394, −5.066031925287915, −4.395300701800299, −4.028798902674945, −3.226091585318893, −2.700175605679140, −1.992254930307600, −0.8789068983733732, −0.5953617488658131,
0.5953617488658131, 0.8789068983733732, 1.992254930307600, 2.700175605679140, 3.226091585318893, 4.028798902674945, 4.395300701800299, 5.066031925287915, 5.423885879718394, 6.170926643113722, 6.671018959377150, 7.347159066727911, 7.552609187442312, 8.149979733027516, 8.781973406613288, 9.277682855229049, 9.811340900223808, 10.42062094762185, 10.88436557636456, 11.34182378042626, 11.62815514193258, 12.36808896008204, 12.63544852750295, 13.05827815559303, 13.60684397945671