| L(s) = 1 | − 2·2-s + 4·4-s − 3·5-s − 7·7-s − 8·8-s + 6·10-s − 23·11-s − 13·13-s + 14·14-s + 16·16-s + 133·17-s + 103·19-s − 12·20-s + 46·22-s + 113·23-s − 116·25-s + 26·26-s − 28·28-s − 141·29-s − 90·31-s − 32·32-s − 266·34-s + 21·35-s − 233·37-s − 206·38-s + 24·40-s − 230·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.268·5-s − 0.377·7-s − 0.353·8-s + 0.189·10-s − 0.630·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 1.89·17-s + 1.24·19-s − 0.134·20-s + 0.445·22-s + 1.02·23-s − 0.927·25-s + 0.196·26-s − 0.188·28-s − 0.902·29-s − 0.521·31-s − 0.176·32-s − 1.34·34-s + 0.101·35-s − 1.03·37-s − 0.879·38-s + 0.0948·40-s − 0.876·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.126728614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.126728614\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| 13 | \( 1 + p T \) |
| good | 5 | \( 1 + 3 T + p^{3} T^{2} \) |
| 11 | \( 1 + 23 T + p^{3} T^{2} \) |
| 17 | \( 1 - 133 T + p^{3} T^{2} \) |
| 19 | \( 1 - 103 T + p^{3} T^{2} \) |
| 23 | \( 1 - 113 T + p^{3} T^{2} \) |
| 29 | \( 1 + 141 T + p^{3} T^{2} \) |
| 31 | \( 1 + 90 T + p^{3} T^{2} \) |
| 37 | \( 1 + 233 T + p^{3} T^{2} \) |
| 41 | \( 1 + 230 T + p^{3} T^{2} \) |
| 43 | \( 1 + 337 T + p^{3} T^{2} \) |
| 47 | \( 1 - 96 T + p^{3} T^{2} \) |
| 53 | \( 1 + 134 T + p^{3} T^{2} \) |
| 59 | \( 1 - 838 T + p^{3} T^{2} \) |
| 61 | \( 1 - 295 T + p^{3} T^{2} \) |
| 67 | \( 1 - 468 T + p^{3} T^{2} \) |
| 71 | \( 1 - 54 T + p^{3} T^{2} \) |
| 73 | \( 1 + 553 T + p^{3} T^{2} \) |
| 79 | \( 1 + 694 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1010 T + p^{3} T^{2} \) |
| 89 | \( 1 - 390 T + p^{3} T^{2} \) |
| 97 | \( 1 + 102 T + p^{3} T^{2} \) |
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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
−9.052932653850584158373293697474, −8.170526922992171580972734470332, −7.46995904113873798005643058109, −6.96758004168410240256244335168, −5.63502053363970023579064403231, −5.22112646668366720339543693097, −3.61918191857082704504753588451, −3.04069392143738865524353185008, −1.70761156404040451926012849659, −0.57125336008409497087876502347,
0.57125336008409497087876502347, 1.70761156404040451926012849659, 3.04069392143738865524353185008, 3.61918191857082704504753588451, 5.22112646668366720339543693097, 5.63502053363970023579064403231, 6.96758004168410240256244335168, 7.46995904113873798005643058109, 8.170526922992171580972734470332, 9.052932653850584158373293697474
