| L(s) = 1 | + 3·3-s + 2·7-s + 6·9-s − 5·13-s + 6·17-s − 6·19-s + 6·21-s − 5·25-s + 9·27-s + 9·29-s + 3·31-s + 8·37-s − 15·39-s + 3·41-s + 8·43-s + 7·47-s − 3·49-s + 18·51-s + 2·53-s − 18·57-s + 4·59-s + 10·61-s + 12·63-s − 8·67-s + 7·71-s + 9·73-s − 15·75-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.755·7-s + 2·9-s − 1.38·13-s + 1.45·17-s − 1.37·19-s + 1.30·21-s − 25-s + 1.73·27-s + 1.67·29-s + 0.538·31-s + 1.31·37-s − 2.40·39-s + 0.468·41-s + 1.21·43-s + 1.02·47-s − 3/7·49-s + 2.52·51-s + 0.274·53-s − 2.38·57-s + 0.520·59-s + 1.28·61-s + 1.51·63-s − 0.977·67-s + 0.830·71-s + 1.05·73-s − 1.73·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.125464991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.125464991\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.136709949264160163400131160245, −7.971762573337755659388498262792, −7.29827869972219621745553541348, −6.35878191326598774890091328679, −5.24164937704147300063994990437, −4.39203650859597269987708926618, −3.82836895032894631923136704271, −2.58998557752273208121368780152, −2.38576533809286741374919255327, −1.13034072062787753744301249711,
1.13034072062787753744301249711, 2.38576533809286741374919255327, 2.58998557752273208121368780152, 3.82836895032894631923136704271, 4.39203650859597269987708926618, 5.24164937704147300063994990437, 6.35878191326598774890091328679, 7.29827869972219621745553541348, 7.971762573337755659388498262792, 8.136709949264160163400131160245
