| L(s) = 1 | + 5-s − 7-s + 4·11-s + 6·13-s − 6·17-s − 4·19-s − 4·23-s + 25-s − 2·29-s − 8·31-s − 35-s − 6·37-s − 6·41-s + 8·43-s + 49-s + 6·53-s + 4·55-s + 4·59-s − 10·61-s + 6·65-s + 8·67-s + 12·71-s − 14·73-s − 4·77-s + 16·79-s + 12·83-s − 6·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s + 1.20·11-s + 1.66·13-s − 1.45·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.169·35-s − 0.986·37-s − 0.937·41-s + 1.21·43-s + 1/7·49-s + 0.824·53-s + 0.539·55-s + 0.520·59-s − 1.28·61-s + 0.744·65-s + 0.977·67-s + 1.42·71-s − 1.63·73-s − 0.455·77-s + 1.80·79-s + 1.31·83-s − 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.203555292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.203555292\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.61436163551664, −15.24464080920357, −14.40649036443695, −14.06314677308826, −13.36980590599996, −13.09846524760841, −12.42555846543627, −11.76794619062268, −11.15766514299013, −10.72488245584010, −10.20960951914631, −9.232440807317811, −9.012938475227982, −8.586534344867953, −7.761583110106062, −6.798662351404793, −6.541382296055403, −5.998334456849256, −5.353425320753953, −4.308643334633613, −3.889989857950881, −3.321500382524666, −2.077367220453996, −1.758476695572186, −0.6052744259921387,
0.6052744259921387, 1.758476695572186, 2.077367220453996, 3.321500382524666, 3.889989857950881, 4.308643334633613, 5.353425320753953, 5.998334456849256, 6.541382296055403, 6.798662351404793, 7.761583110106062, 8.586534344867953, 9.012938475227982, 9.232440807317811, 10.20960951914631, 10.72488245584010, 11.15766514299013, 11.76794619062268, 12.42555846543627, 13.09846524760841, 13.36980590599996, 14.06314677308826, 14.40649036443695, 15.24464080920357, 15.61436163551664
