| L(s) = 1 | − 2·4-s + 5-s + 7-s + 4·16-s − 3·17-s − 6·19-s − 2·20-s + 8·23-s − 4·25-s − 2·28-s − 29-s + 6·31-s + 35-s − 2·37-s + 6·41-s − 43-s − 3·47-s + 49-s − 12·53-s − 7·59-s − 12·61-s − 8·64-s − 11·67-s + 6·68-s − 8·71-s + 4·73-s + 12·76-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s + 0.377·7-s + 16-s − 0.727·17-s − 1.37·19-s − 0.447·20-s + 1.66·23-s − 4/5·25-s − 0.377·28-s − 0.185·29-s + 1.07·31-s + 0.169·35-s − 0.328·37-s + 0.937·41-s − 0.152·43-s − 0.437·47-s + 1/7·49-s − 1.64·53-s − 0.911·59-s − 1.53·61-s − 64-s − 1.34·67-s + 0.727·68-s − 0.949·71-s + 0.468·73-s + 1.37·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9704782002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9704782002\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.00283931496337, −12.59778736598729, −12.24403318025890, −11.39850879946205, −11.16375123449828, −10.51833158741271, −10.23655361578726, −9.600395055209412, −9.099376269568967, −8.842649899855113, −8.414280992063337, −7.682796550344675, −7.499912129036841, −6.575320195385122, −6.199347567223076, −5.823439137797749, −4.961182711572739, −4.703512534532918, −4.375720824787013, −3.628086952122563, −3.046171889867294, −2.436335207342124, −1.700046983861318, −1.202055219482138, −0.2882332332078963,
0.2882332332078963, 1.202055219482138, 1.700046983861318, 2.436335207342124, 3.046171889867294, 3.628086952122563, 4.375720824787013, 4.703512534532918, 4.961182711572739, 5.823439137797749, 6.199347567223076, 6.575320195385122, 7.499912129036841, 7.682796550344675, 8.414280992063337, 8.842649899855113, 9.099376269568967, 9.600395055209412, 10.23655361578726, 10.51833158741271, 11.16375123449828, 11.39850879946205, 12.24403318025890, 12.59778736598729, 13.00283931496337
