| L(s) = 1 | − 2·7-s − 3·9-s − 6·11-s + 2·13-s + 6·17-s + 2·19-s − 6·23-s − 6·29-s + 4·31-s + 6·37-s − 2·41-s + 4·43-s + 10·47-s − 3·49-s + 2·53-s − 10·59-s + 10·61-s + 6·63-s − 4·67-s + 16·71-s + 6·73-s + 12·77-s + 9·81-s − 8·83-s + 6·89-s − 4·91-s − 2·97-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 9-s − 1.80·11-s + 0.554·13-s + 1.45·17-s + 0.458·19-s − 1.25·23-s − 1.11·29-s + 0.718·31-s + 0.986·37-s − 0.312·41-s + 0.609·43-s + 1.45·47-s − 3/7·49-s + 0.274·53-s − 1.30·59-s + 1.28·61-s + 0.755·63-s − 0.488·67-s + 1.89·71-s + 0.702·73-s + 1.36·77-s + 81-s − 0.878·83-s + 0.635·89-s − 0.419·91-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.124598414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.124598414\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.515619767221866917709938130636, −7.905215376884215396313485465394, −7.40802331669392530547209574659, −6.09541777505772482967493011895, −5.77043351526859900745946994682, −5.02719603997497771198726075842, −3.74549304830489392851332240527, −3.05274044832240915515195000909, −2.27346656460613455309222123602, −0.60684224882274745382116979714,
0.60684224882274745382116979714, 2.27346656460613455309222123602, 3.05274044832240915515195000909, 3.74549304830489392851332240527, 5.02719603997497771198726075842, 5.77043351526859900745946994682, 6.09541777505772482967493011895, 7.40802331669392530547209574659, 7.905215376884215396313485465394, 8.515619767221866917709938130636
