L(s) = 1 | + 5.02·2-s + 17.2·4-s + 5·5-s + 46.5·8-s + 25.1·10-s + 0.888·11-s − 25.9·13-s + 95.6·16-s + 96.3·17-s − 89.0·19-s + 86.2·20-s + 4.46·22-s + 116.·23-s + 25·25-s − 130.·26-s + 222.·29-s − 12.9·31-s + 108.·32-s + 483.·34-s + 91.2·37-s − 447.·38-s + 232.·40-s + 98.4·41-s + 392.·43-s + 15.3·44-s + 585.·46-s + 220.·47-s + ⋯ |
L(s) = 1 | + 1.77·2-s + 2.15·4-s + 0.447·5-s + 2.05·8-s + 0.794·10-s + 0.0243·11-s − 0.553·13-s + 1.49·16-s + 1.37·17-s − 1.07·19-s + 0.964·20-s + 0.0432·22-s + 1.05·23-s + 0.200·25-s − 0.983·26-s + 1.42·29-s − 0.0748·31-s + 0.600·32-s + 2.44·34-s + 0.405·37-s − 1.91·38-s + 0.919·40-s + 0.374·41-s + 1.39·43-s + 0.0525·44-s + 1.87·46-s + 0.683·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(8.552756770\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.552756770\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 5.02T + 8T^{2} \) |
| 11 | \( 1 - 0.888T + 1.33e3T^{2} \) |
| 13 | \( 1 + 25.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 96.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 89.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 116.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 222.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 12.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 91.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 98.4T + 6.89e4T^{2} \) |
| 43 | \( 1 - 392.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 220.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 228.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 13.5T + 2.05e5T^{2} \) |
| 61 | \( 1 - 205.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 325.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 583.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 950.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 451.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 164.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 884.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 62.1T + 9.12e5T^{2} \) |
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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
−8.601757916538448742638715500492, −7.56330323277182013915655238235, −6.86102769200083666831633887865, −6.09018890353227476352897142689, −5.44304587000889713746456498017, −4.71444468899495033503642268457, −3.95098051026681906349955230434, −2.93471060463403888453952998932, −2.33448684078510403649306676377, −1.05066349423197204254979750577,
1.05066349423197204254979750577, 2.33448684078510403649306676377, 2.93471060463403888453952998932, 3.95098051026681906349955230434, 4.71444468899495033503642268457, 5.44304587000889713746456498017, 6.09018890353227476352897142689, 6.86102769200083666831633887865, 7.56330323277182013915655238235, 8.601757916538448742638715500492