L(s) = 1 | − 4·5-s − 2·7-s − 11-s + 2·13-s − 2·17-s + 6·19-s − 4·23-s + 11·25-s − 6·29-s + 4·31-s + 8·35-s + 6·37-s + 10·41-s − 6·43-s + 8·47-s − 3·49-s + 4·55-s + 4·59-s + 6·61-s − 8·65-s − 8·67-s − 2·73-s + 2·77-s − 10·79-s + 12·83-s + 8·85-s − 4·91-s + ⋯ |
L(s) = 1 | − 1.78·5-s − 0.755·7-s − 0.301·11-s + 0.554·13-s − 0.485·17-s + 1.37·19-s − 0.834·23-s + 11/5·25-s − 1.11·29-s + 0.718·31-s + 1.35·35-s + 0.986·37-s + 1.56·41-s − 0.914·43-s + 1.16·47-s − 3/7·49-s + 0.539·55-s + 0.520·59-s + 0.768·61-s − 0.992·65-s − 0.977·67-s − 0.234·73-s + 0.227·77-s − 1.12·79-s + 1.31·83-s + 0.867·85-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−7.67320533894418133408984131383, −7.18248964476174506488801568304, −6.34054091500064994507355487148, −5.57823930343146157243564654217, −4.58513645502667514689217314045, −3.92629622327595901120260263865, −3.37349693704824660584249894341, −2.56064825858819520078795119937, −0.999211074167011910715018735120, 0,
0.999211074167011910715018735120, 2.56064825858819520078795119937, 3.37349693704824660584249894341, 3.92629622327595901120260263865, 4.58513645502667514689217314045, 5.57823930343146157243564654217, 6.34054091500064994507355487148, 7.18248964476174506488801568304, 7.67320533894418133408984131383