L(s) = 1 | + 5-s − 7-s + 6·13-s − 2·17-s − 8·19-s + 8·23-s + 25-s − 2·29-s − 4·31-s − 35-s + 2·37-s + 6·41-s + 4·43-s + 8·47-s + 49-s + 10·53-s − 4·59-s + 2·61-s + 6·65-s + 4·67-s − 12·71-s − 2·73-s − 8·79-s + 4·83-s − 2·85-s + 6·89-s − 6·91-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s + 1.66·13-s − 0.485·17-s − 1.83·19-s + 1.66·23-s + 1/5·25-s − 0.371·29-s − 0.718·31-s − 0.169·35-s + 0.328·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 1.37·53-s − 0.520·59-s + 0.256·61-s + 0.744·65-s + 0.488·67-s − 1.42·71-s − 0.234·73-s − 0.900·79-s + 0.439·83-s − 0.216·85-s + 0.635·89-s − 0.628·91-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
\(L(1)\) |
\(\approx\) |
\(2.393744848\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.393744848\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.57678487169175, −15.13522704984469, −14.64903602111867, −13.92673635353632, −13.37024571490891, −12.86806787934590, −12.74162580859021, −11.71415470487318, −11.02630184924767, −10.76519329389449, −10.28718143404568, −9.267601898669318, −8.948992614974241, −8.585584672639214, −7.723518881056564, −6.942974054356073, −6.502571673048224, −5.863828128449019, −5.415885834161694, −4.294595334923704, −4.030161285366436, −3.070503431014483, −2.391166227931310, −1.541426529768578, −0.6562282347406039,
0.6562282347406039, 1.541426529768578, 2.391166227931310, 3.070503431014483, 4.030161285366436, 4.294595334923704, 5.415885834161694, 5.863828128449019, 6.502571673048224, 6.942974054356073, 7.723518881056564, 8.585584672639214, 8.948992614974241, 9.267601898669318, 10.28718143404568, 10.76519329389449, 11.02630184924767, 11.71415470487318, 12.74162580859021, 12.86806787934590, 13.37024571490891, 13.92673635353632, 14.64903602111867, 15.13522704984469, 15.57678487169175