L(s) = 1 | + 2-s − 3-s + 4-s + 3.93·5-s − 6-s + 8-s + 9-s + 3.93·10-s + 1.27·11-s − 12-s + 5.94·13-s − 3.93·15-s + 16-s − 3.05·17-s + 18-s − 19-s + 3.93·20-s + 1.27·22-s − 0.148·23-s − 24-s + 10.5·25-s + 5.94·26-s − 27-s − 6.60·29-s − 3.93·30-s + 8.01·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.76·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 1.24·10-s + 0.383·11-s − 0.288·12-s + 1.64·13-s − 1.01·15-s + 0.250·16-s − 0.740·17-s + 0.235·18-s − 0.229·19-s + 0.880·20-s + 0.271·22-s − 0.0310·23-s − 0.204·24-s + 2.10·25-s + 1.16·26-s − 0.192·27-s − 1.22·29-s − 0.719·30-s + 1.44·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.283450466\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.283450466\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 - 3.93T + 5T^{2} \) |
| 11 | \( 1 - 1.27T + 11T^{2} \) |
| 13 | \( 1 - 5.94T + 13T^{2} \) |
| 17 | \( 1 + 3.05T + 17T^{2} \) |
| 23 | \( 1 + 0.148T + 23T^{2} \) |
| 29 | \( 1 + 6.60T + 29T^{2} \) |
| 31 | \( 1 - 8.01T + 31T^{2} \) |
| 37 | \( 1 - 6.14T + 37T^{2} \) |
| 41 | \( 1 + 6.28T + 41T^{2} \) |
| 43 | \( 1 - 7.91T + 43T^{2} \) |
| 47 | \( 1 + 8.22T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 1.84T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 - 2.61T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 - 7.44T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 5.40T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.176256037765246561894008510279, −7.01345045321421939443711865931, −6.31233742386885707510547386055, −6.09100723667226712286217531816, −5.42585634707671471683740865986, −4.60001760294172369051777638794, −3.81920295516404994240928973801, −2.75115161154838930910488409880, −1.85726818912877732063659177921, −1.13021298474506601424669403145,
1.13021298474506601424669403145, 1.85726818912877732063659177921, 2.75115161154838930910488409880, 3.81920295516404994240928973801, 4.60001760294172369051777638794, 5.42585634707671471683740865986, 6.09100723667226712286217531816, 6.31233742386885707510547386055, 7.01345045321421939443711865931, 8.176256037765246561894008510279