L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 5.12·7-s − 8-s + 9-s + 5.12·11-s + 12-s − 2·13-s − 5.12·14-s + 16-s − 7.12·17-s − 18-s + 4·19-s + 5.12·21-s − 5.12·22-s − 23-s − 24-s + 2·26-s + 27-s + 5.12·28-s + 2·29-s − 32-s + 5.12·33-s + 7.12·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s + 1.93·7-s − 0.353·8-s + 0.333·9-s + 1.54·11-s + 0.288·12-s − 0.554·13-s − 1.36·14-s + 0.250·16-s − 1.72·17-s − 0.235·18-s + 0.917·19-s + 1.11·21-s − 1.09·22-s − 0.208·23-s − 0.204·24-s + 0.392·26-s + 0.192·27-s + 0.968·28-s + 0.371·29-s − 0.176·32-s + 0.891·33-s + 1.22·34-s + 0.166·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.371946192\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.371946192\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - 5.12T + 7T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 7.12T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 - 0.876T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 5.12T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 3.12T + 89T^{2} \) |
| 97 | \( 1 + 0.246T + 97T^{2} \) |
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
−8.678501056620397074524127119769, −7.949294618535340611291231523243, −7.32297255166330895441215356457, −6.65183061270178398082841946814, −5.60644202218822576847615714532, −4.51933200358358999783950809664, −4.13173782140712780330682031828, −2.68274395935091927829555649873, −1.86071573285890450348112535710, −1.08867540938457529309456509133,
1.08867540938457529309456509133, 1.86071573285890450348112535710, 2.68274395935091927829555649873, 4.13173782140712780330682031828, 4.51933200358358999783950809664, 5.60644202218822576847615714532, 6.65183061270178398082841946814, 7.32297255166330895441215356457, 7.949294618535340611291231523243, 8.678501056620397074524127119769