| L(s) = 1 | − 3·3-s + 12.7·5-s + 6.52·7-s + 9·9-s + 28.3·13-s − 38.3·15-s − 17.9·17-s + 65.4·19-s − 19.5·21-s + 185·23-s + 38.7·25-s − 27·27-s − 20.9·29-s − 184.·31-s + 83.5·35-s + 142.·37-s − 85.0·39-s + 79.6·41-s + 487.·43-s + 115.·45-s − 334.·47-s − 300.·49-s + 53.9·51-s + 187.·53-s − 196.·57-s + 312.·59-s + 646.·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.14·5-s + 0.352·7-s + 0.333·9-s + 0.604·13-s − 0.660·15-s − 0.256·17-s + 0.790·19-s − 0.203·21-s + 1.67·23-s + 0.310·25-s − 0.192·27-s − 0.133·29-s − 1.06·31-s + 0.403·35-s + 0.631·37-s − 0.349·39-s + 0.303·41-s + 1.72·43-s + 0.381·45-s − 1.03·47-s − 0.875·49-s + 0.148·51-s + 0.485·53-s − 0.456·57-s + 0.690·59-s + 1.35·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.614248550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.614248550\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 12.7T + 125T^{2} \) |
| 7 | \( 1 - 6.52T + 343T^{2} \) |
| 13 | \( 1 - 28.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 17.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 65.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 185T + 1.21e4T^{2} \) |
| 29 | \( 1 + 20.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 184.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 142.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 79.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 487.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 334.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 187.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 312.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 646.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 83.5T + 3.00e5T^{2} \) |
| 71 | \( 1 - 361.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 188.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 649.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 63.0T + 7.04e5T^{2} \) |
| 97 | \( 1 - 168.T + 9.12e5T^{2} \) |
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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
−9.349109898536325890686152029289, −8.460472815770720358533139461729, −7.36520934323996481686487526202, −6.63918291619117237134780530210, −5.69473278944831157142513847098, −5.27435971309695587928125290169, −4.18573280480337221507914266848, −2.93326285198049774160489134658, −1.77881662531722912543037571579, −0.868975440952638944319607524285,
0.868975440952638944319607524285, 1.77881662531722912543037571579, 2.93326285198049774160489134658, 4.18573280480337221507914266848, 5.27435971309695587928125290169, 5.69473278944831157142513847098, 6.63918291619117237134780530210, 7.36520934323996481686487526202, 8.460472815770720358533139461729, 9.349109898536325890686152029289
