L(s) = 1 | − 3-s + 5-s + 9-s − 11-s − 15-s + 2·17-s + 6·19-s + 8·23-s + 25-s − 27-s − 4·31-s + 33-s − 2·37-s − 8·41-s + 8·43-s + 45-s + 12·47-s − 2·51-s − 10·53-s − 55-s − 6·57-s + 12·59-s + 2·61-s − 4·67-s − 8·69-s − 8·71-s + 12·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.258·15-s + 0.485·17-s + 1.37·19-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.718·31-s + 0.174·33-s − 0.328·37-s − 1.24·41-s + 1.21·43-s + 0.149·45-s + 1.75·47-s − 0.280·51-s − 1.37·53-s − 0.134·55-s − 0.794·57-s + 1.56·59-s + 0.256·61-s − 0.488·67-s − 0.963·69-s − 0.949·71-s + 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.346070474\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.346070474\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 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
−15.06179161384163, −14.42444841314774, −14.07118421602284, −13.23057810187980, −13.11930929825416, −12.29760098507285, −11.95407279241655, −11.23000964697779, −10.85789421855554, −10.23226244854298, −9.767232642924492, −9.106603007195371, −8.744887373592634, −7.774873507287412, −7.329468684810624, −6.863834698593887, −6.098871521444175, −5.437950093653248, −5.204108575132612, −4.482914763864828, −3.552609617529418, −3.040442563069127, −2.196081598006722, −1.291657716718942, −0.6641632439699440,
0.6641632439699440, 1.291657716718942, 2.196081598006722, 3.040442563069127, 3.552609617529418, 4.482914763864828, 5.204108575132612, 5.437950093653248, 6.098871521444175, 6.863834698593887, 7.329468684810624, 7.774873507287412, 8.744887373592634, 9.106603007195371, 9.767232642924492, 10.23226244854298, 10.85789421855554, 11.23000964697779, 11.95407279241655, 12.29760098507285, 13.11930929825416, 13.23057810187980, 14.07118421602284, 14.42444841314774, 15.06179161384163