L(s) = 1 | − 7-s − 4·11-s + 4·13-s + 2·17-s + 6·19-s − 8·23-s − 5·25-s + 2·29-s − 4·31-s − 10·37-s + 10·41-s − 4·43-s − 4·47-s + 49-s − 2·53-s + 10·59-s + 8·61-s + 8·67-s − 6·73-s + 4·77-s − 16·79-s + 2·83-s − 18·89-s − 4·91-s − 2·97-s − 4·103-s − 16·107-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 1.20·11-s + 1.10·13-s + 0.485·17-s + 1.37·19-s − 1.66·23-s − 25-s + 0.371·29-s − 0.718·31-s − 1.64·37-s + 1.56·41-s − 0.609·43-s − 0.583·47-s + 1/7·49-s − 0.274·53-s + 1.30·59-s + 1.02·61-s + 0.977·67-s − 0.702·73-s + 0.455·77-s − 1.80·79-s + 0.219·83-s − 1.90·89-s − 0.419·91-s − 0.203·97-s − 0.394·103-s − 1.54·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−8.059305277600964705263533268485, −7.46590261373315505042250951052, −6.60140796065074920648863068053, −5.59485565479475106439435856224, −5.44324662131016484771078155754, −4.07721841167186091289402411806, −3.46756796989567776692371205688, −2.52828773073271220396016108760, −1.41777100868395989536837747433, 0,
1.41777100868395989536837747433, 2.52828773073271220396016108760, 3.46756796989567776692371205688, 4.07721841167186091289402411806, 5.44324662131016484771078155754, 5.59485565479475106439435856224, 6.60140796065074920648863068053, 7.46590261373315505042250951052, 8.059305277600964705263533268485