| L(s) = 1 | − 2.05·2-s + 2.23·4-s + 1.05·5-s − 7-s − 0.492·8-s − 2.18·10-s + 3.80·13-s + 2.05·14-s − 3.46·16-s + 2.56·17-s − 0.180·19-s + 2.37·20-s − 0.433·23-s − 3.87·25-s − 7.83·26-s − 2.23·28-s + 9.03·29-s − 0.492·31-s + 8.11·32-s − 5.28·34-s − 1.05·35-s − 0.775·37-s + 0.371·38-s − 0.521·40-s + 3.09·41-s − 1.61·43-s + 0.893·46-s + ⋯ |
| L(s) = 1 | − 1.45·2-s + 1.11·4-s + 0.473·5-s − 0.377·7-s − 0.174·8-s − 0.689·10-s + 1.05·13-s + 0.550·14-s − 0.866·16-s + 0.622·17-s − 0.0413·19-s + 0.530·20-s − 0.0904·23-s − 0.775·25-s − 1.53·26-s − 0.423·28-s + 1.67·29-s − 0.0884·31-s + 1.43·32-s − 0.906·34-s − 0.178·35-s − 0.127·37-s + 0.0602·38-s − 0.0825·40-s + 0.483·41-s − 0.246·43-s + 0.131·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.033037088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.033037088\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 + 2.05T + 2T^{2} \) |
| 5 | \( 1 - 1.05T + 5T^{2} \) |
| 13 | \( 1 - 3.80T + 13T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 + 0.180T + 19T^{2} \) |
| 23 | \( 1 + 0.433T + 23T^{2} \) |
| 29 | \( 1 - 9.03T + 29T^{2} \) |
| 31 | \( 1 + 0.492T + 31T^{2} \) |
| 37 | \( 1 + 0.775T + 37T^{2} \) |
| 41 | \( 1 - 3.09T + 41T^{2} \) |
| 43 | \( 1 + 1.61T + 43T^{2} \) |
| 47 | \( 1 + 0.502T + 47T^{2} \) |
| 53 | \( 1 - 9.94T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 - 5.93T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 4.27T + 73T^{2} \) |
| 79 | \( 1 - 1.37T + 79T^{2} \) |
| 83 | \( 1 - 7.47T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−7.951666934288750887533799207075, −7.46747726494006141008100768106, −6.48353804257680522625669559950, −6.17734506355066650121435962333, −5.21022517842721892741453919909, −4.24908275847505138178801115853, −3.33598831039095057394173645111, −2.38544360101993884856538465336, −1.48779132614420047696544846698, −0.68153743558677321703945468756,
0.68153743558677321703945468756, 1.48779132614420047696544846698, 2.38544360101993884856538465336, 3.33598831039095057394173645111, 4.24908275847505138178801115853, 5.21022517842721892741453919909, 6.17734506355066650121435962333, 6.48353804257680522625669559950, 7.46747726494006141008100768106, 7.951666934288750887533799207075
