| L(s) = 1 | − 2·5-s − 3·9-s + 6·13-s − 2·17-s − 25-s + 10·29-s + 2·37-s − 10·41-s + 6·45-s − 14·53-s − 10·61-s − 12·65-s + 6·73-s + 9·81-s + 4·85-s − 10·89-s − 18·97-s − 2·101-s − 6·109-s − 14·113-s − 18·117-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 9-s + 1.66·13-s − 0.485·17-s − 1/5·25-s + 1.85·29-s + 0.328·37-s − 1.56·41-s + 0.894·45-s − 1.92·53-s − 1.28·61-s − 1.48·65-s + 0.702·73-s + 81-s + 0.433·85-s − 1.05·89-s − 1.82·97-s − 0.199·101-s − 0.574·109-s − 1.31·113-s − 1.66·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.345357893193109848205355184463, −7.81663575443705708449207741292, −6.62808933612086279625365823883, −6.21664603313252085616146347555, −5.21491067832233438164314315088, −4.31344318158954137705495075068, −3.51581292263084701318927160501, −2.79060940321229954934356841383, −1.36200050793739601416696080977, 0,
1.36200050793739601416696080977, 2.79060940321229954934356841383, 3.51581292263084701318927160501, 4.31344318158954137705495075068, 5.21491067832233438164314315088, 6.21664603313252085616146347555, 6.62808933612086279625365823883, 7.81663575443705708449207741292, 8.345357893193109848205355184463
