L(s) = 1 | − 2-s − 3-s + 4-s + 4.29·5-s + 6-s − 4.35·7-s − 8-s + 9-s − 4.29·10-s + 1.15·11-s − 12-s + 4.35·14-s − 4.29·15-s + 16-s + 0.493·17-s − 18-s + 1.78·19-s + 4.29·20-s + 4.35·21-s − 1.15·22-s + 3.38·23-s + 24-s + 13.4·25-s − 27-s − 4.35·28-s + 6.93·29-s + 4.29·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.92·5-s + 0.408·6-s − 1.64·7-s − 0.353·8-s + 0.333·9-s − 1.35·10-s + 0.349·11-s − 0.288·12-s + 1.16·14-s − 1.10·15-s + 0.250·16-s + 0.119·17-s − 0.235·18-s + 0.408·19-s + 0.960·20-s + 0.950·21-s − 0.247·22-s + 0.705·23-s + 0.204·24-s + 2.69·25-s − 0.192·27-s − 0.823·28-s + 1.28·29-s + 0.784·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.173304842\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.173304842\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 4.29T + 5T^{2} \) |
| 7 | \( 1 + 4.35T + 7T^{2} \) |
| 11 | \( 1 - 1.15T + 11T^{2} \) |
| 17 | \( 1 - 0.493T + 17T^{2} \) |
| 19 | \( 1 - 1.78T + 19T^{2} \) |
| 23 | \( 1 - 3.38T + 23T^{2} \) |
| 29 | \( 1 - 6.93T + 29T^{2} \) |
| 31 | \( 1 + 2.22T + 31T^{2} \) |
| 37 | \( 1 + 3.87T + 37T^{2} \) |
| 41 | \( 1 + 6.31T + 41T^{2} \) |
| 43 | \( 1 - 7.38T + 43T^{2} \) |
| 47 | \( 1 + 1.78T + 47T^{2} \) |
| 53 | \( 1 + 2.51T + 53T^{2} \) |
| 59 | \( 1 - 6.63T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 4.09T + 67T^{2} \) |
| 71 | \( 1 - 9.38T + 71T^{2} \) |
| 73 | \( 1 - 0.374T + 73T^{2} \) |
| 79 | \( 1 - 2.65T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 - 0.835T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−10.03842649393895429826362242363, −9.333588600467977876199985707333, −8.709557879791102178032278926234, −7.05162526198095272053958533698, −6.56565537594463246814404323788, −5.92898083242957195711713899712, −5.11195033686480993975713344555, −3.34439808579787990288029354792, −2.32503462239410640587673454972, −0.988857246942512103753552122256,
0.988857246942512103753552122256, 2.32503462239410640587673454972, 3.34439808579787990288029354792, 5.11195033686480993975713344555, 5.92898083242957195711713899712, 6.56565537594463246814404323788, 7.05162526198095272053958533698, 8.709557879791102178032278926234, 9.333588600467977876199985707333, 10.03842649393895429826362242363