| L(s) = 1 | − 5-s − 2·13-s − 6·17-s + 4·19-s − 6·23-s + 25-s − 6·29-s + 4·31-s + 2·37-s + 6·41-s − 10·43-s − 6·47-s + 6·53-s + 12·59-s − 2·61-s + 2·65-s + 2·67-s + 12·71-s − 2·73-s + 8·79-s + 6·83-s + 6·85-s − 6·89-s − 4·95-s − 2·97-s + 6·101-s − 14·103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.554·13-s − 1.45·17-s + 0.917·19-s − 1.25·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.328·37-s + 0.937·41-s − 1.52·43-s − 0.875·47-s + 0.824·53-s + 1.56·59-s − 0.256·61-s + 0.248·65-s + 0.244·67-s + 1.42·71-s − 0.234·73-s + 0.900·79-s + 0.658·83-s + 0.650·85-s − 0.635·89-s − 0.410·95-s − 0.203·97-s + 0.597·101-s − 1.37·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.232658369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.232658369\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.84705746007080707747306683679, −7.02154360198695631694990895041, −6.53418433426898175298075758337, −5.64205532362515499365982008548, −4.94497854065074162838342241506, −4.20748366665210762546679649438, −3.57122359553730834302448832528, −2.57999868391564120109847897545, −1.85149647197241415984585908916, −0.52100658180081998819004041300,
0.52100658180081998819004041300, 1.85149647197241415984585908916, 2.57999868391564120109847897545, 3.57122359553730834302448832528, 4.20748366665210762546679649438, 4.94497854065074162838342241506, 5.64205532362515499365982008548, 6.53418433426898175298075758337, 7.02154360198695631694990895041, 7.84705746007080707747306683679
