L(s) = 1 | − 3-s + 9-s − 4·11-s − 4·13-s + 4·17-s + 2·19-s − 6·23-s − 5·25-s − 27-s + 2·31-s + 4·33-s + 4·39-s − 10·41-s + 2·43-s + 8·47-s − 7·49-s − 4·51-s + 6·53-s − 2·57-s − 10·59-s − 4·61-s + 4·67-s + 6·69-s + 14·73-s + 5·75-s − 10·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s − 1.10·13-s + 0.970·17-s + 0.458·19-s − 1.25·23-s − 25-s − 0.192·27-s + 0.359·31-s + 0.696·33-s + 0.640·39-s − 1.56·41-s + 0.304·43-s + 1.16·47-s − 49-s − 0.560·51-s + 0.824·53-s − 0.264·57-s − 1.30·59-s − 0.512·61-s + 0.488·67-s + 0.722·69-s + 1.63·73-s + 0.577·75-s − 1.12·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5956968059\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5956968059\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 37 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.20157467817065, −14.41258133172894, −13.95781245888032, −13.49093404748781, −12.79501508210131, −12.29356092831784, −11.91673075434345, −11.45474547499335, −10.59367627557968, −10.26638631725906, −9.765696163410394, −9.363283979939025, −8.225368420628290, −8.003957941725453, −7.381376993122076, −6.881043530728000, −5.983649343075017, −5.577250685529744, −5.073687748220470, −4.436606504550731, −3.685272228561993, −2.903514627491931, −2.242953071914407, −1.435381214319975, −0.2988153270683272,
0.2988153270683272, 1.435381214319975, 2.242953071914407, 2.903514627491931, 3.685272228561993, 4.436606504550731, 5.073687748220470, 5.577250685529744, 5.983649343075017, 6.881043530728000, 7.381376993122076, 8.003957941725453, 8.225368420628290, 9.363283979939025, 9.765696163410394, 10.26638631725906, 10.59367627557968, 11.45474547499335, 11.91673075434345, 12.29356092831784, 12.79501508210131, 13.49093404748781, 13.95781245888032, 14.41258133172894, 15.20157467817065