L(s) = 1 | − 12.2i·5-s + 14.2·7-s − 104. i·11-s + 75.2·13-s − 341. i·17-s + 706.·19-s − 596. i·23-s + 474.·25-s + 1.30e3i·29-s − 1.02e3·31-s − 175. i·35-s + 563.·37-s − 99.1i·41-s + 896.·43-s − 430. i·47-s + ⋯ |
L(s) = 1 | − 0.491i·5-s + 0.291·7-s − 0.860i·11-s + 0.445·13-s − 1.18i·17-s + 1.95·19-s − 1.12i·23-s + 0.758·25-s + 1.54i·29-s − 1.07·31-s − 0.143i·35-s + 0.411·37-s − 0.0589i·41-s + 0.484·43-s − 0.194i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.421156482\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.421156482\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 12.2iT - 625T^{2} \) |
| 7 | \( 1 - 14.2T + 2.40e3T^{2} \) |
| 11 | \( 1 + 104. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 75.2T + 2.85e4T^{2} \) |
| 17 | \( 1 + 341. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 706.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 596. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.30e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.02e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 563.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 99.1iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 896.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 430. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 5.27e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 5.63e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 1.13e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 1.35e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 5.68e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.23e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 6.13e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 7.50e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 8.72e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 5.44e3T + 8.85e7T^{2} \) |
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
−9.049747521394862466781773457512, −8.119574759650668694513473590671, −7.34693958794132735436081983891, −6.43793444177363218725212014305, −5.32171383978194777182165963671, −4.91061062405111062542685699737, −3.57581662478201854313184469686, −2.79804712779173639960879011496, −1.31522420639132507973579204862, −0.56209821737177858137407758456,
1.08293445756007305172834738861, 2.06043606666157125440888431493, 3.25727928781784306332830589183, 4.06963931122092026385245872032, 5.19766032021286874244052450540, 5.94619851986537433857302650238, 6.98661541683233997335892136963, 7.61947229703703482544866363001, 8.394951446901511007236113552636, 9.542194211542448743804626459258