| L(s) = 1 | − 4·7-s + 2·11-s + 13-s + 2·17-s + 6·19-s + 6·23-s − 2·29-s + 10·31-s + 2·37-s + 6·41-s + 10·43-s − 4·47-s + 9·49-s + 2·53-s + 6·59-s + 2·61-s − 4·67-s + 6·71-s + 6·73-s − 8·77-s + 12·79-s + 16·83-s − 2·89-s − 4·91-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.603·11-s + 0.277·13-s + 0.485·17-s + 1.37·19-s + 1.25·23-s − 0.371·29-s + 1.79·31-s + 0.328·37-s + 0.937·41-s + 1.52·43-s − 0.583·47-s + 9/7·49-s + 0.274·53-s + 0.781·59-s + 0.256·61-s − 0.488·67-s + 0.712·71-s + 0.702·73-s − 0.911·77-s + 1.35·79-s + 1.75·83-s − 0.211·89-s − 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.779693579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.779693579\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−14.59124060964456, −13.97928308574504, −13.56430096754948, −13.07427821308572, −12.55729813126915, −12.08133515137741, −11.57251268915096, −10.98163673073443, −10.36600374469497, −9.781313690191153, −9.366200695391879, −9.088100644230147, −8.267811101713356, −7.596681498515287, −7.136322987599784, −6.423982895480456, −6.189583568931414, −5.430499435777477, −4.840529060325049, −3.984330429578789, −3.465718418053423, −2.953488803322256, −2.333735242322442, −1.036188489426497, −0.7529727091431540,
0.7529727091431540, 1.036188489426497, 2.333735242322442, 2.953488803322256, 3.465718418053423, 3.984330429578789, 4.840529060325049, 5.430499435777477, 6.189583568931414, 6.423982895480456, 7.136322987599784, 7.596681498515287, 8.267811101713356, 9.088100644230147, 9.366200695391879, 9.781313690191153, 10.36600374469497, 10.98163673073443, 11.57251268915096, 12.08133515137741, 12.55729813126915, 13.07427821308572, 13.56430096754948, 13.97928308574504, 14.59124060964456
